Skip main navigation

New lower prices! Get up to 50% off 1000s of courses. 

Explore courses

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
This article is from the free online

Introduction to Mathematical Methods for University-Level Science

Created by
FutureLearn - Learning For Life

Reach your personal and professional goals

Unlock access to hundreds of expert online courses and degrees from top universities and educators to gain accredited qualifications and professional CV-building certificates.

Join over 18 million learners to launch, switch or build upon your career, all at your own pace, across a wide range of topic areas.

Start Learning now