What are logarithms?

In science there are many situations where we have a graph that is of the following form rather than linear. In this article, logarithms will be introduced as a tool to solve exponential growth questions. This topic will allow you to work with results of this form in science.

Figure 1: The graph of (y=3^{x}).

This graph represents exponential growth, and we see that the value of (y) increases very quickly.

Alternatively we may have a graph of the following form

Figure 2: The graph of (y=3^{-2x}).

This graph instead shows an example of exponential decay. We see that the value of (y) quickly decreases and seems to be approaching 0.

Finding the inverse of (a^{x})

We may want to solve an equation of the form (a^{x}=b)

To be able to solve an equation of this form we must introduce the logarithm.

Definition

To solve the equation (a^{x}=b). we denote (a) to be the base and (x) to be the exponent, then

(log_{a}(b)=x).

Note: On a standard calculator (log(x)=log_{10}(x)).

An example

Solve (2^{x}=64).

We use the logarithm so that

(x=log_{2}(64)) when we type this on a calculator we find (x=6).

An example

Solve (10times 9^{x}=17.32.)

We first rewrite the equation so that

(9^{x}=1.732).

Then we can find (x) by writing

[x=log_{9}(1.732)=0.2500]

The inverse of (e^{x})

An example

Solve the equation (e^{x}=5)

We use the natural logarithm so that

[x=ln(5)=1.6094]

An example

Solve the equation (e^{-3x}=0.2231)

First use the natural logarithm so that

[-3x=ln(0.2231)]

Then we solve to find (x) so that

[x=frac{ln(0.2231)}{-3}=0.5000]

Can you think of an example in your area of science of exponential growth or decay?