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Graphs of Exponents and Logarithms

Exponential graphs are used to describe a function known as . This function is special because its derivative is exactly the same so .

What does the graph of look like?

There are some clear observations we can make about the exponential graph.

The graph of

• We can see the graph tends to infinity, meaning that it is unbounded.

• We can see the y-intercept is 1. This is because .

• We can see it is asymptotic to the x-axis. This is because has no solutions.

• e has a value of around 2.7.

How can we transform an exponential graph?

Standard transformation rules apply to exponential functions – check out Transformations of Graphs for more information.

Let's look at an example.

Sketch the graph of

.

What does a logarithmic graph look like?

Again, there are some clear observations we can make.

This is a general graph.

• This is asymptotic to the y-axis as the value of a logarithm of base 10 of 0 does not exist.

• The x-intercept is 1 as

• The graph still tends to infinity but much slower.

How can we transform logarithmic graphs?

Again, we use the standard rules for transformations.

Let's look at an example of this.

A painting increases in value with respect to time. The relationship is stated by . Sketch the graph When does it start to have a positive value?

What does the graph of the natural logarithm look like?

Let's look at the graph and make some observations.

This is very similar to the logarithmic graph, with no y-intercept and an x-intercept of 1 as .

One distinct thing we can see here by inverse functions is that ln is the inverse of e. So if then .

The inverse of a function in a cartesian sense is a reflection on the line . So is a reflection of on the line .

Graph of exponential, natural logarithm and

Let's look at an example question about the graph of the natural logarithm.

By using a graphical method, find the value of lny when and .

Graphs of Exponents and Logarithms - Key takeaways

• Exponential graphs are in the form .

• Exponential and Logarithmic graphs exhibit standard properties of transformations.

• They are also differentiable.

• Logarithmic graphs are very similar.

• They are useful for estimating values.

• Linear progressions can be shown by mapping out against .

• The natural logarithmic graph is just the exponential graph reflected on the line

Logarithmic graphs don’t tend to infinity like exponential graphs.

They are both asymptotic to different values.

Logarithmic graphs are exponential graphs reflected on the line y=x.

They can be used to model pricing and their properties can help when modelling populations such as of bacteria and infected populations.

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