# 5.04 Logarithms

Lesson

## Introduction

We've seen equations like y=B^x before. It's straightforward enough to find y when we know x, but is it possible to find x if we know y?

The expression B^x, if x is a natural number, means the number of B factors multiplied together is x. So to find x in 3^x=81 we ask how many 3 factors are in 81, and the answer is 4. But we saw from exponential graphs that x can in general be any real number, including irrational numbers. In that case it doesn't make sense to multiply B \, x times.

## Logarithms

Logarithms are expressions of the form \log_{B}y, where B is some number and y is a pronumeral. B is called the base of the logarithm. The definition of a logarithm is that if y=B^x then \log_{B}y=x.

In other words, \log_{B}y is the number of B factors that multiply together to make y. It follows that \log_{3}81=4, since 3^4=81.

Of course, the value of the logarithm could be any real number. We will soon see how to find the exact values of logarithms, but we can approximate the value using a calculator.

First note that by convention, if B is not specified that means it has a base of 10. So \log_y=\log_{10}y.

If we wanted to find \log_{}81, then we can press the "log" button on a calculator and then enter 81. This gives us 1.908 to three decimal places.

### Examples

#### Example 1

Rewrite the equation 9^x=81 in logarithmic form (with the index as the subject of the equation).

Worked Solution
Create a strategy

Use the definition: 'If y=B^x then \log_{B}y=x.'

Apply the idea

From the given equation we have B=9 and y=81. So the logarithmic form of 9^x=81 is:\log_{9}81=x

#### Example 2

Evaluate \log_{8} \left(\dfrac{1}{64}\right).

Worked Solution
Create a strategy

Use the definition: 'If y=B^x then \log_{B}y=x.'

Apply the idea

Let x be the index or the subject of the equation. So we have:

#### Example 3

Evaluate \log_{10}45. Round your answer to two decimal places.

Worked Solution
Create a strategy

Use a calculator or other technology to find a decimal value for the logarithm.

Apply the idea

Press the log button and enter the number 45 to get:\log_{10}45 \approx 1.65

Idea summary

Logarithms are expressions of the form \log_{B}y, where B is any number and y is a pronumeral.

In \log_{B}y, \, B, is the base of the logarithm.

By convention, if the base is not specified then B=10.

If y=B^x then \log_{B}y=x, so y is the number of B factors that are multiplied together to give x.

### Outcomes

#### VCMNA356 (10a)

Use the definition of a logarithm to establish and apply the laws of logarithms and investigate logarithmic scales in measurement