Learn9MA0Paper 2 · Pure Mathematics 2Exponentials and logarithmsLaws of logarithms

Exponentials and logarithms · Subtopic 2 of 5

Laws of Logarithms

What a logarithm is (the power you need), the three laws that come straight from the index laws, and how to combine and simplify logarithms, including $\ln$ and $\lg$ .

Cambridge 9709 Paper 2Cambridge 9709 Paper 3Edexcel YMA01 WMA12Edexcel 9MA0 Paper 1Edexcel 9MA0 Paper 2AQA 7357 Paper 1≈ 40 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Convert between $a^x=b$ and $\log_a b=x$ .
Evaluate simple logarithms without a calculator.
Use the laws $\log(xy)$ , $\log\left(\tfrac xy\right)$ and $\log(x^k)$ to combine and split logarithms.
Use $\ln e=1$ , $\ln1=0$ , $e^{\ln x}=x$ and $\ln(e^x)=x$ .

Pause and think

$2^?=32$ . $10^?=1000$ . $5^?=\tfrac{1}{25}$ . Find each missing power.

Section 2 of 14

Core idea

Start hereTurning multiplication into addition

Before calculators, astronomers spent months multiplying huge numbers. In 1614 John Napier published a trick: replace each number by its logarithm (the power it is of a fixed base), then multiplying numbers becomes adding logarithms. Logarithms are simply powers, written the other way round.

The big idea

$\log_ab=c$ means $a^c=b$ : a log is an exponent. Because index laws add powers when multiplying, logs obey $\log(xy)=\log x+\log y$ , $\log\frac xy=\log x-\log y$ , $\log x^k=k\log x$ .

1. Definition

\log_ab=c\iff a^c=b

$a$: the base: $a>0$ , $a\neq1$ .
$b$: the number: must be $>0$ .
$c$: the logarithm: the power.

Exponential and logarithm undo each other

2^3=8

and

\log_28=3

are the same fact; the graphs are reflections in

y=x

$\log_28=3$ since $2^3=8$ .
$\log_{10}0.01=-2$ since $10^{-2}=0.01$ .
$\log_a1=0$ , $\log_aa=1$ , $\log_aa^k=k$ .

2. The laws, proved

Product law

$\text{Let }\log_ax=p,\ \log_ay=q$
Name the logs.
$x=a^p,\ y=a^q$
Definition.
$xy=a^pa^q=a^{p+q}$
Index law.
$\log_a(xy)=p+q=\log_ax+\log_ay$
Definition again.

Quotient and power laws

$\frac xy=a^{p-q}\Rightarrow\log_a\frac xy=\log_ax-\log_ay$
Same method, dividing.
$x^k=(a^p)^k=a^{kp}\Rightarrow\log_ax^k=k\log_ax$
Power of a power.

Common misconceptionLaws that do not exist

\log(x+y)\neq\log x+\log y

;

\frac{\log x}{\log y}\neq\log\frac xy

;

(\log x)^2\neq2\log x

Worked exampleSimplify $2\log_33+\log_312-\log_34$

Write as a single number.

Goal: A number.
Strategy: Power law, then combine.

$\log_39+\log_312-\log_34$
$2\log_33=\log_39$ .
$=\log_3\frac{9\times12}4=\log_327$
Product and quotient.
$=3$
$3^3=27$ .

Answer

3

✓ CheckNumerically:

2+2.262-1.262=3

✓.

3. Change of base

\log_ab=\dfrac{\log_cb}{\log_ca}

$\log_ab=x\Rightarrow a^x=b$
Definition.
$\log_c(a^x)=\log_cb$
Take $\log_c$ of both sides.
$x\log_ca=\log_cb\Rightarrow x=\frac{\log_cb}{\log_ca}$
Power law; divide.

\log_57=\frac{\ln7}{\ln5}=1.209

Pause and think

Simplify $\log_48$ .

Summary

Log = exponent.
Product → sum; quotient → difference; power → multiple.
Change of base.

Logarithm: The power the base must be raised to.

HistoryNapier’s 20-year project. John Napier, a Scottish laird, published the first table of logarithms in 1614 after about twenty years of calculation. Astronomers were overjoyed: multiplying huge numbers became adding their logarithms. Henry Briggs then produced base-10 tables, which were used for over 350 years until electronic calculators arrived.

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