$\begin{align}(i)&\frac{1}{8} =\frac{1}{2^3}=2^{-3}\\[5pt] \end{align}$	$\begin{align}(ii)&\frac{1}{100} =\frac{1}{10^2}=10^{-2}\\[5pt] \end{align}$	$\begin{align}(iii)&\frac{1}{125} =\frac{1}{5^3}=5^{-3}\\[5pt] \end{align}$
$\begin{align}(iv)&\frac{1}{81} =\frac{1}{3^4}=3^{-4}\\[5pt] \end{align}$	$\begin{align}(v)&0.01=\frac{1}{100} =\frac{1}{10^2}=10^{-2}\\[5pt] \end{align}$	$\begin{align}(vi)&0.001=\frac{1}{1000} =\frac{1}{10^3}=10^{-3}\\[5pt] \end{align}$
$\begin{align}(i)&a^2\times a^3 =a^{2+3}=a^5\\[5pt] \end{align}$	$\begin{align}(ii)&x^5\times x =x^{5+1}=x^6\\[5pt] \end{align}$	$\begin{align}(iii)&\frac{x^5 \times x^7}{x^{11}}=\frac{x^{12}}{x^{11}}=x^{12-11}=x\\[5pt] \end{align}$
$\begin{align}(iv)&\frac{a^3 \times a^5}{a^2\times a^6}=\frac{a^8}{a^8}=a^0=1\\[5pt] \end{align}$	$\begin{align}(v)&\frac{p^3\times p^{-1}}{p} =\frac {p^2}{p^1}=p\\[5pt] \end{align}$	$\begin{align}(vi)&\frac{x^0\times x^5}{x} =\frac {x^5}{x^1}={x^4}\\[5pt] \end{align}$
$\begin{align}(i)&2^2\times 2^3 =2^5\\[5pt] &\underline{\underline{32}} \end{align}$	$\begin{align}(ii)&\frac {3^7}{3^4} =3^3\\[5pt] &\underline{\underline{27}} \end{align}$	$\begin{align}(iii)&\frac {3^2\times 3^8}{3^5} =\frac {3^{10}}{3^5}=3^5\\[5pt] &\underline{\underline{243}} \end{align}$
$\begin{align}(iv)&\frac {5^3\times 5^0}{5} =\frac {5^3}{5^1}=5^2\\[5pt] &\underline{\underline{25}} \end{align}$	$\begin{align}(v)&\frac {10^2\times 10^3}{10\times 10^4} =\frac {10^5}{10^5}=10^0\\[5pt] &\underline{\underline{1}} \end{align}$	$\begin{align}(vi)&\frac {2^5\times 2^3}{2^6\times 2^2} =\frac {2^8}{2^8}=2^0\\[5pt] &\underline{\underline{1}} \end{align}$

By studying this lesson you will be able to

Simplify numerical expressions using logarithms.