StudySmarter - The all-in-one study app.

4.8 • +11k Ratings

More than 3 Million Downloads

Free

Logarithmic Functions

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Lagrange Error Bound
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
- Linear Functions
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
- P Series
- Particle Model Motion
- Particular Solutions to Differential Equations
- Polar Coordinates
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
- Vectors in Space
- Washer Method
- Decision Maths
- Geometry
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Altitude
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
- Area of Circular Sector
- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
- Geometric Probability
- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isometry
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
- Surface Area of Cylinder
- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Law of Gravitation
- Newton's Second Law
- Newton's Third Law
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Conic Sections
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Roots of Unity
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Order Recurrence Relation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Bias in Experiments
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Chi Square Test for Goodness of Fit
- Chi Square Test for Homogeneity
- Chi Square Test for Independence
- Chi-Square Distribution
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
- Conditional Probability
- Conducting a Study
- Conducting a Survey
- Conducting an Experiment
- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
- Confidence Interval for Slope of Regression Line
- Confidence Interval for the Difference of Two Means
- Confidence Intervals
- Correlation Math
- Cumulative Frequency
- Data Analysis
- Data Interpretation
- Discrete Random Variable
- Distributions
- Dot Plot
- Empirical Rule
- Errors in Hypothesis Testing
- Estimator Bias
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Test of Two Population Proportions
- Hypothesis Testing
- Inference for Distributions of Categorical Data
- Inferences in Statistics
- Large Data Set
- Least Squares Linear Regression
- Linear Interpolation
- Linear Regression
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Normal Distribution Percentile
- Point Estimation
- Probability
- Probability Calculations
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Quartiles
- Random Variables
- Randomized Block Design
- Residual Sum of Squares
- Residuals
- Sample Mean
- Sample Proportion
- Sampling
- Sampling Distribution
- Scatter Graphs
- Single Variable Data
- Skewness
- Standard Deviation
- Standard Normal Distribution
- Statistical Graphs
- Statistical Measures
- Stem and Leaf Graph
- Sum of Independent Random Variables
- Survey Bias
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
- Type II Error
- Types of Data in Statistics
- Venn Diagrams

If you want to know who can play their music the loudest, you can look at the decibel level of the sound systems, which is measured using logarithms. We will look at the definition of a logarithmic function, how to graph logarithmic functions, and the rules for using them.

The logarithmic function is defined to be the inverse of the exponential function. See Inverse Functions for more details on exactly how functions and their inverses are related, but in short two functions *f* and *g* are inverses of each other if

For more information on how functions and their inverses are related, see Inverse Functions .

When you look at the graphs of an exponential function, and the corresponding logarithmic function, they are reflections of each other across the line . In other words, if the point is on one of the graphs, then the point is on the other graph.

You will be using the **rules of logarithms**:

- Product Rule:
- Quotient Rule:
- Reciprocal Rule:
- Power Rule:
- Proportional Rule (Change of Base formula):

Simplify the expression .

**Answer: **

**Step 1:** If that were a logarithm base 10 then the answer would be using properties of inverse functions. So the idea is to use the Proportion Rule (also known as the change of base formula) to make it into a base 10 logarithm first. There are two ways to think about doing this, and you get the same answer either way.

The first way to think about it is to use the fact that 10 is the number you are raising to a power to get:

The second way is to look at the logarithm and see that it is base 100, and use that to get:

and then solve for to get that Both methods work, and you can use the one that is easiest for you to understand and remember.

So you have

**Step 2:** Now using properties of exponents,

So the expression completely simplified is

Whenever you use the rules of logarithms, you need to be sure that you use values for x that make sense for the function, as well as the exponential function, since they are inverses.

For example, you can't try and use negative values for in because the exponential function is always positive. A negative x value would make negative (x and y switch with inverses).You also can't use a negative constant for the base in a logarithm because you can't use it as the base of an exponential function.

A logarithmic function is any function of the form where , , and . This is read *f(x)* equals the log base *b* of *x*.

When there is no base *b* listed, it is taken to be 10. So means the same thing as .

What is the inverse of the function ?

Answer:

Remember that when no base is listed it is taken to be 10. So the inverse of is .

What is the inverse of the function ?

Answer:

Remember that inverses work both ways! So the inverse of is .

List at least 3 points on the graph of without graphing the function or using a calculator.

**Answer:**

This problem looks trickier than it actually is. You already know that the inverse of is , and that if is a point on the graph of then is a point on the graph of .

Step 1: Exponential functions have a y intercept at , so the point is on the graph of .

Step 2: To get two more points on the graph, evaluate points on the graph of . Choosing two random values, .

Step 3: Evaluate another point on the graph of

Step 4: So the points and are on the graph of . So you know that the points and are points on the graph of .

So without using a calculator, you can see that three points on the graph of are , , and .

Looking at the key takeaway, because the two are equivalent, you can use properties of the exponential function (see Exponential Functions ) when thinking about properties of logarithmic functions. An exponential function can't have a negative number for the base, which is why the base of the logarithmic function can't be negative either. The exponential function only takes on positive values for *y*, so the logarithmic function only can use positive numbers for *x*.

This leads to the following **properties of logarithmic functions**:

- the domain is
- the range is
- there is no
*y-intercept* - the
*x-intercept*is at - the vertical asymptote has the equation

First let's look at some examples graphed together to see how the base *b* affects the graph.

Here

They all

- have the same vertical asymptote at .
- have the same
*x*intercept at . - are concave down
- are increasing functions.

Let's take a look at the base of the functions in the example and use the change of base formula.

So really they are all just constant multiples of .

What if instead, the base was a fractional power of 2? For example if but now you have

, , and , then

which means that again they are constant multiples of , but they should be flipped over the *x* axis as well.

logarithms with fractions as the base | StudySmarter Originals

So as you can see, these three new functions are

- decreasing,
- concave up,
- have as the vertical asymptote, and
- all have the same
*x*intercept at .

Logarithmic functions are used to model things like noise and the intensity of earthquakes. Let's take a look at some real-life examples in action!

Sounds are measured on a logarithmic scale using the unit, *decibels (dB)*. Sound can be modeled using the equation:

Where

- is the power of the sound,
- is the smallest sound a person can hear, and
- is the number of decibels for the power :

Say you are thinking of buying a new speaker. Speaker A says it has a noise rating of 50 decibels, while speaker B says it has a noise rating of 75 decibels. How much more intense is the sound from speaker B than from speaker A?

**Answer:**

Step 1: For comparison, call the decibel level of speaker A, and the decibel level of speaker B. From the information given, you know that:

- speaker A has where is the power of speaker A
- speaker B has where is the power of speaker B

Step 2: Taking the equation for speaker A and writing it in terms of will let you substitute it into the equation for speaker B. So

Step 3: Substituting this into the equation for speaker B,

So, the sound from speaker B is about 316 times more intense than that of speaker A!

Earthquakes are measured on a logarithmic scale called the Richter scale. The magnitude of an earthquake is a measure of how much energy is released. Here is amplitude of the smallest wave that a seismograph (the device that measures how much the earth is moving) can measure. Then the formula for the Richter scale measurement of an earthquake is

where measures the amplitude of the earthquake wave. In general an earthquake measures between 2 and 10 on the Richter scale. Ones that scores less than 5 on the scale are considered relatively minor, and anything above an 8 on the scale is likely to cause quite a bit of damage. In fact, a magnitude 5 earthquake is 10 times as powerful as a magnitude 4 earthquake.

Suppose that an earthquake in Indiana had a magnitude of 8.1 on the Richter scale, but one on the same day in California was 1.26 times as intense. What was the magnitude of the earthquake in California?

**Answer:**

Step 1: Using the definition of the Richter scale, and using for the amplitude of the Indiana earthquake, the earthquake in Indiana had

Step 2: Making each side to the base 10 and solving for :

Now you can use the fact that the California earthquake was 1.26 times as intense as the Indiana one, or in other words, if is the amplitude of the California earthquake, then . So

That means the earthquake in California measured about 8.2 on the Richter scale.

The derivative of the logarithmic function is

For information on the derivatives of logarithmic functions, see Derivative of the Logarithmic Function.

- is equivalent to
- the formula for a logarithmic function is where , and .
- is the same thing as
- logarithms are used in measuring things like decibels and how strong earthquakes are

The logarithmic function is the inverse of the exponential function.

You don't solve logarithmic functions, you solve logarithmic equations.

More about Logarithmic Functions

Be perfectly prepared on time with an individual plan.

Test your knowledge with gamified quizzes.

Create and find flashcards in record time.

Create beautiful notes faster than ever before.

Have all your study materials in one place.

Upload unlimited documents and save them online.

Identify your study strength and weaknesses.

Set individual study goals and earn points reaching them.

Stop procrastinating with our study reminders.

Earn points, unlock badges and level up while studying.

Create flashcards in notes completely automatically.

Create the most beautiful study materials using our templates.

Sign up to highlight and take notes. It’s 100% free.