State the quotient law of logarithms.

\loga\left(\frac xy\right)=\logax-\logay.

Is \loga(x+y)=\logax+\logay?

No. The product law applies to xy, not x+y.

Laws of logarithmsEdexcel International A Level Maths: Flashcards

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$\log_ax=y$ means what in index form?

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$\log_ax=y$ means what in index form?: $a^y=x$ .
State the product law of logarithms.: $\log_a(xy)=\log_ax+\log_ay$ .
State the quotient law of logarithms.: $\log_a\left(\frac xy\right)=\log_ax-\log_ay$ .
State the power law of logarithms.: $\log_a\left(x^k\right)=k\log_ax$ .
What is $\log_a\left(\frac1x\right)$ ?: $-\log_ax$ .
What is $\log_aa$ ?: $1$ .
Conditions on $a$ , $x$ and $y$ for the laws?: $a>0$ , $a\neq1$ and $x,y>0$ .
Write $\log_a\sqrt x$ using the power law.: $\frac12\log_ax$ .
Is $\log_a(x+y)=\log_ax+\log_ay$ ?: No. The product law applies to $xy$ , not $x+y$ .
Is $\log_a(x^2)=(\log_ax)^2$ ?: No. $\log_a(x^2)=2\log_ax$ .
Write $2\log_a5-\log_a4$ as a single logarithm.: $\log_a\frac{25}{4}$ .
Given $\log_a2=p$ , find $\log_a8$ .: $3p$ .
What is $\log_2 8+\log_2 4$ as a single number?: $3+2=5$ , which is $\log_232$ .

Exam questions on Laws of logarithms

Given that $\log_a 2=p$ and $\log_a 5=q$ , where $a$ is a positive constant and $a\neq1$ .
Express $\log_a\left(\frac{25}{8}\right)$ in terms of $p$ and $q$ .2 marks
$x$ and $y$ are positive real numbers and $a$ is a positive constant with $a\neq1$ . $\log_a x=3$ and $\log_a y=5$ .
Find the value of $\log_a\left(\frac1{x^2y}\right)$ .2 marks
Throughout this question $a$ is a positive constant with $a\neq1$ . Do not use a calculator.
Show that $\log_a48-\log_a3=4\log_a2$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).