Motivation

A brief refresher

To jog your memory, we recall some basic definitions and rules for manipulating exponentials and logarithms. For further details, we refer to the module Indices and logarithms (Years 9–10).

Logarithms and exponentials are inverse operations. In particular, for \(a>1\),

\[ x = a^y \ \iff \ y = \log_a x. \]

The following index laws hold for any bases \(a,b>0\) and any real numbers \(m\) and \(n\):

\begin{alignat*}{2} a^m a^n &= a^{m+n} &\qquad\qquad\qquad \dfrac{a^m}{a^n} &= a^{m-n}\\ (a^m)^n &= a^{mn} & (ab)^m &= a^m b^m\\ \Bigl(\dfrac{a}{b}\Bigr)^m &= \dfrac{a^m}{b^m}. \end{alignat*}

Some simple consequences of the index laws are, for \(a>0\) and a positive integer \(n\):

\[ a^0 = 1 \qquad\qquad\qquad a^{\frac{1}{n}} = \sqrt[n]{a} \qquad\qquad\qquad a^{-1} = \dfrac{1}{a}. \]

The following logarithm laws hold for any base \(a>1\), any positive \(x\) and \(y\), and any real number \(n\):

\begin{alignat*}{2} \log_a 1 &= 0 &\qquad\qquad\qquad \log_a a &= 1\\ \log_a (xy) &= \log_a x + \log_a y & \log_a \dfrac{x}{y} &= \log_a x - \log_a y\\ \log_a \dfrac{1}{x} &= -\log_a x & \log_a (x^n) &= n \, \log_a x. \end{alignat*}

Also, recall the change of base formula:

\[ \log_b x = \dfrac{ \log_a x }{ \log_a b }, \] for any \(a,b > 1\) and any positive \(x\).

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