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What are logarithms?

Logarithms are a tool to solve exponential growth questions. This article will allow you to work with results of this form in science.
Using Logarithms To Solve Equations
© University of Nottingham

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.

A graph which has an x-axis that goes from 0 to 10 and a y-axis that goes from 0 to just over 60,000. The title is “y equals 3 to the power of x.” As x increases, the value of y-increases. The gradient of the graph is increasing as the value of x increases. 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

A graph which has an x-axis that goes from 0 to 10 and a y-axis that goes from 0 to 9. The title is “y equals 3 to the power of -2x.” As x increases, the value of y-decreases. The graph represents exponential decay and we see that as x increases the value of y is getting closer to 0. The value of y is decreasing at a faster rate initially, but as x increases the value of y is still decreasing, but at a slower rate. 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})

The natural logarithm has the base of the constant (e) where (e=2.718….). We label the natural logarithm as (ln(x)) so that (log_{e}(x)=ln(x)).
We can use the (ln) function to solve equations as in the following two examples.
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?

© University of Nottingham
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Introduction to Mathematical Methods for University-Level Science

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