A logarithm (usually abbreviated to ‘log’) is the inverse of an exponential function. It is a new topic at A Level, building on what you learnt at GCSE about powers and indices. It’s important to get a thorough introduction to logarithms – you’ll be building on this as you move onto logarithmic graphs and logarithmic equations.

If you want to see the introduction to logarithms below in PDF or presentable format, with a wider range of questions and answers, click here. For a multiple-choice quiz practicing the required prior knowledge, click here. If you think you’ve mastered logarithms, try these logarithm games or these exam-style questions.

As an example of an exponential:

10^{3} = 1000

The same statement written as a logarithm is:

log_{10} 1000 = 3

The small number next to ‘log’ is the base – this is the power we are working with. This example would be said as ‘log to base 10 of 1000 equals 3’. The left side of the statement above is saying ‘To what power do I need to raise 10, to get 1000?’, the answer to which is 3.

In more general terms:

If 𝑎^{𝑐}= 𝑏 then log_{𝑎}𝑏 = 𝑐

Note that a logarithm is only defined if the base (in this case, *a*) is a positive, real number not equal to 1.

**Example Question 1**

Rewrite the following using logarithms:

**a. 5 ^{2} = 25**

This question is dealing with powers of 5, so the base is 5. What power of 5 is needed to make 25:

log_{5} 25 = 2

**b. 6 ^{0} = 1**

log_{6} 1 = 0

**Example Question 2**

Rewrite the following using indices:

**a. log _{9} 81 = 2**

Here, the base is 9. The statement is saying that 9 needs to be raised to the power of 2 to get 81:

9^{2} = 81

**b. **

In some cases, you can calculate logs mentally. For example, if you’re asked to find log_{2} 32, you should be able to work that out by asking yourself what power of 2 gives 32. The answer is 5, as 2^{5} = 32, therefore log_{2} 32 = 5.

**Example Question 3** (non-calculator)

Evaluate:

**a. log _{3} 81**

3^{4} = 81 therefore log_{3} 81 = 4

**b. log _{16} 2**

= 2, so log_{16} 2 =

However, solving logarithms mentally is not always possible. If you’re asked to find log_{2} 17, you are unlikely to be able to find a solution mentally, because 17 is not an integer power of 2.

That’s where your calculator comes in. To find log2 17, use the button, being careful to put 2 and 17 in the correct places. This will give 4.09 (2d.p).

**Example Question 4** (calculator)

Evaluate the following. Give you answers correct to 3s.f.:

**a. log _{3} 21**

log3 21 = 2.77

**b. **

There are three log laws which you’ll need to know and apply; they’re similar to the laws of indices. The laws are:

Test them out. You could let *a* = 10, *b* = 1000 and *c* = 100 in the first law if you want numbers that you can work with mentally, or you could use any positive numbers you like if you have a calculator to hand.

**Example Question 5**

Write each question as a single logarithm

**a. log _{3} 9 + log_{3} 81**

= log_{3} (9 x 81)

= log_{3} 729

**b. log _{a} 15 – log_{a} 90**

= log_{a} (15 ÷ 90)

= log_{a}

**c. 2log _{3} 9 + log_{3} 11**

= log_{3} (9^{2}) + log_{3} 11

= log_{3} (81 x 11)

= log_{3} 891

**Summary**

log_{𝑎}𝑏 = 𝑐 is equivalent to 𝑎^{𝑐}= 𝑏 Where 𝑎 is a positive, real number not equal to 1. log_{𝑎}𝑏 + log_{𝑎}𝑐 = log_{𝑎}𝑏𝑐 log_{𝑎}𝑏 - log_{𝑎}𝑐 = log_{𝑎}log_{𝑎}(𝑏𝑐) = clog_{𝑎}𝑏 To evaluate a logarithm on your calculator, use the key.

### Introduction to Logarithms – **Practice Questions**

**Non-Calculator**

1. Complete the table, using equivalent statements:

2. For each question, find the value of 𝑥:

a. log_{𝑥} 9 = 2

b. log_{6} 𝑥 = -2

c. log_{11} 1331 = 𝑥

d. log_{20} 400 = 𝑥

e. log_{4𝑥} 16 = 4

f. log_{2} 8𝑥 = 6

3. Express each as a single logarithm:

a. log_{3} 8 + log_{3} 2

b. 2log_{2} 𝑎 + log_{2} 𝑏

c. log_{𝑦} 8 – 2log_{𝑦} 4

d. log_{2} 2 – log_{2} 0.5 – log_{2} 8

e. 4log𝑝 𝑚 – log𝑝 2𝑚

f. log_{10 }3 + 0.5log_{10} 9 – log_{10} 19

4. Given that log_{𝑡} 𝑎 = 𝑥 and log_{𝑡} 𝑏 = 𝑦, express log_{𝑡} (𝑎^{2}𝑏^{3}) in terms of 𝑥 and 𝑦.

5. 𝑎 = log_{𝑐} 2 and 𝑏 = log_{𝑐} 3. Express log_{𝑐} 48 in terms of 𝑎 and 𝑏.

**Calculator**

6. Giving your answers correct to 2 s.f., evaluate:

a. log_{7} 111

b. log_{0.3} 5

c. log_{6 }9

d. log_{0.1} 0.38

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