# Logarithms in Base 10

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}

$$\log_{10}{10^x} = x$$
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}