\loga(a^x) and a^{\logax}?

Both give x (inverse functions).

Is \log(x+y)=\log x+\log y?

No. The laws apply to products and quotients, not sums.

How do you solve a^x=b?

Take logarithms: x\ln a=\ln b, so x=\frac{\ln b}{\ln a}.

Logarithms and their lawsEdexcel A-Level Maths: Flashcards

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$\log_ax=y$ means?

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$\log_ax=y$ means?: $a^y=x$
$\log_aa$ and $\log_a1$ ?: $1$ and $0$
$\log_a(a^x)$ and $a^{\log_ax}$ ?: Both give $x$ (inverse functions).
Domain of $\log_ax$ ?: $x>0$
What is $\ln x$ ?: $\log_ex$ , the inverse of $e^x$
Solve $e^{ax+b}=p$ .: $ax+b=\ln p$
Solve $\ln(ax+b)=q$ .: $ax+b=e^q$
Addition law?: $\log_ax+\log_ay=\log_a(xy)$
Subtraction law?: $\log_ax-\log_ay=\log_a\frac xy$
Power law?: $k\log_ax=\log_a(x^k)$
$-\log_ax$ as a single logarithm?: $\log_a\frac1x$
$\log_a\sqrt x$ in terms of $\log_ax$ ?: $\frac12\log_ax$
Is $\log(x+y)=\log x+\log y$ ?: No. The laws apply to products and quotients, not sums.
How do you solve $a^x=b$ ?: Take logarithms: $x\ln a=\ln b$ , so $x=\frac{\ln b}{\ln a}$ .

Exam questions on Logarithms and their laws

Consider the logarithms $\log_5125$ and $\log_2\frac18$ .
Find the value of $\log_48$ .2 marks
Let $p=\log_a2$ and $q=\log_a5$ , where $a>1$ .
Express $\log_a\sqrt{20}$ in terms of $p$ and $q$ .2 marks
The value $\pounds V$ of one savings account after $t$ years is $V=2000e^{0.04t}$ . A second account has value $\pounds W=1500\times1.05^t$ after $t$ years.
Find how many years it takes for the first account to reach $\pounds3000$ , giving your answer to 3 significant figures.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).