The logarithm to base e; the inverse of e^x

e^{\ln x} and \ln(e^x)?

Both equal x (for e^{\ln x}, x0)

Subtraction law of logarithms?

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

-\logax as a single log?

\loga\left(\frac1x\right) (power law with k=-1)

-\frac12\logax as a single log?

\loga\left(\frac{1}{\sqrt x}\right) (power law with k=-\frac12)

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

No. The sum of logs is \loga(xy); there is no law for \loga(x+y)

After solving a log equation, what must you check?

That each answer keeps every logarithm's argument positive

Logarithms and their lawsAQA A-Level Maths: Flashcards

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Question

Definition of $\log_ax$?

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Definition of $\log_ax$ ?: $\log_ax=y\iff a^y=x$ (for $a>0$ , $x>0$ )
$\log_a1$ and $\log_aa$ ?: $0$ and $1$
For which $x$ is $\log_ax$ defined?: Only $x>0$
What is $\ln x$ ?: The logarithm to base $e$ ; the inverse of $e^x$
$e^{\ln x}$ and $\ln(e^x)$ ?: Both equal $x$ (for $e^{\ln x}$ , $x>0$ )
Addition law of logarithms?: $\log_ax+\log_ay=\log_a(xy)$
Subtraction law of logarithms?: $\log_ax-\log_ay=\log_a\left(\frac xy\right)$
Power law of logarithms?: $k\log_ax=\log_a(x^k)$
$-\log_ax$ as a single log?: $\log_a\left(\frac1x\right)$ (power law with $k=-1$ )
$-\frac12\log_ax$ as a single log?: $\log_a\left(\frac{1}{\sqrt x}\right)$ (power law with $k=-\frac12$ )
Solve $\ln x=3$ .: $x=e^3$
Is $\log_a(x+y)=\log_ax+\log_ay$ ?: No. The sum of logs is $\log_a(xy)$ ; there is no law for $\log_a(x+y)$
Value of $\log_28$ and of $\log_2\frac18$ ?: $3$ and $-3$
After solving a log equation, what must you check?: That each answer keeps every logarithm's argument positive

Exam questions on Logarithms and their laws

In this question, evaluate each logarithm without using a calculator.
Find the exact value of $\log_48$ .2 marks
Let $a=\log_{10}2$ and $b=\log_{10}3$ .
Write $\log_{10}\sqrt{18}$ in terms of $a$ and $b$ .2 marks
The positive real numbers $x$ and $y$ are such that $\ln x=p$ and $\ln y=q$ .
Express $\ln\left(\dfrac{x^3}{\sqrt y}\right)$ in terms of $p$ and $q$ .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).