Many positive numbers can be easily written in the form of $10^x$.
$$ \begin{align}
10\ 000 &= 10^4 \\
1000 &= 10^3 \\
100 &= 10^2 \\
10 &= 10^1 \\
1 &= 10^0 \\
0.1 &= 10^{-1} \\
0.01 &= 10^{-2} \\
0.001 &= 10^{-3}
\end{align}$$
All positive numbers can be written in the form $10^x$ by using logarithms in base $10$.
$$y = 10^x \rightarrow \log_{10}{y} = x$$
Therefore the exponential forms (index forms) can be written in logarithmic forms;
$$ \begin{align}
10\ 000 = 10^4 &\rightarrow \log_{10}{10\ 000} = 4 \\
1000 = 10^3 &\rightarrow \log_{10}{1000} = 3 \\
100 = 10^2 &\rightarrow \log_{10}{100} = 2 \\
10 = 10^1 &\rightarrow \log_{10}{10} = 1 \\
1 = 10^0 &\rightarrow \log_{10}{1} = 0 \\
0.1 = 10^{-1} &\rightarrow \log_{10}{0.1} = -1 \\
0.01 = 10^{-2} &\rightarrow \log_{10}{0.01} = -2 \\
0.001 = 10^{-3} &\rightarrow \log_{10}{0.001} = -3
\end{align}$$
These expressions can be re-written as follows;
$$ \begin{align}
\log_{10}{10\ 000} = 4 &\rightarrow \log_{10}{10^4} = 4 \\
\log_{10}{1000} = 3 &\rightarrow \log_{10}{10^3} = 3 \\
\log_{10}{100} = 2 &\rightarrow \log_{10}{10^2} = 2 \\
\log_{10}{10} = 1 &\rightarrow \log_{10}{10^1} = 1 \\
\log_{10}{1} = 0 &\rightarrow \log_{10}{10^0} = 0 \\
\log_{10}{0.1} = -1 &\rightarrow \log_{10}{10^{-1}} = -1 \\
\log_{10}{0.01} = -2 &\rightarrow \log_{10}{10^{-2}} = -2 \\
\log_{10}{0.001} = -3 &\rightarrow \log_{10}{10^{-3}} = -3
\end{align}$$
These lead to a pattern:
$$\log_{10}{10^x} = x$$
Often $\log_{10}{y}$ can be written in a simpler form $\log{y}$.
$$\log_{10}{10^x} = \log{10^x} = x$$
Example 1
Without using a calculator, find $\log_{10}{100\ 000}$.
\( \begin{align}
\log_{10}{100\ 000} &= \log_{10}{10^5} \\
&= 5
\end{align} \)
Example 2
Find $\log_{10}{0.0001}$ without using a calculator.
\( \begin{align}
\log_{10}{0.0001} &= \log_{10}{10^{-4}} \\
&= -4
\end{align} \)
Example 3
Without using a calculator, find $\log_{10}{\sqrt{10}}$.
\( \begin{align}
\log_{10}{\sqrt{10}} &= \log_{10}{10^{\frac{1}{2}}} \\
&= \dfrac{1}{2}
\end{align} \)
Example 4
Without using a calculator, find $\log_{10}{\sqrt[3]{100}}$.
\( \begin{align}
\log_{10}{\sqrt[3]{100}} &= \log_{10}{\sqrt[3]{10^2}} \\
&= \log_{10}{10^{\frac{2}{3}}} \\
&= \dfrac{2}{3}
\end{align} \)
Example 5
Write $8$ in the form $10^x$ where $x$ is correct to 4 significant figures.
\( \begin{align}
10^x &= 8 \\
x &= \log_{10}{8} \\
&= 0.9031 &\text{by calculator}\\
\therefore 8 &= 10^{0.9031}
\end{align} \)
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