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.
\( \begin{align}
10^x &= 8 \\
x &= \log_{10}{8} \\
&= 0.9031 &\text{by calculator}\\
\therefore 8 &= 10^{0.9031}
\end{align} \)
Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorials Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume