Logarithmic Functions: Learn It 3

Common Logarithms

Sometimes we may see a logarithm written without a base. In this case, we assume that the base is 10. In other words, the expressions [latex]\text{log}(x)[/latex] means [latex]\text{log}_{10}(x).[/latex] We call a base-[latex]10[/latex] logarithm a common logarithm. Common logarithms are used to measure the Richter Scale of earthquakes. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.

common logarithm

A common logarithm is a logarithm with base [latex]10[/latex]. We write [latex]\text{log}_{10}(x)[/latex] simply as [latex]\text{log}(x)[/latex]. The common logarithm of a positive number [latex]x[/latex] satisfies the following definition:

 

For [latex]x>0[/latex],

[latex]y=\text{log}(x)\text{ is equivalent to }{10}^{y}=x[/latex]

 

We read [latex]\text{log}(x)[/latex] as, “the logarithm with base [latex]10[/latex] of [latex]x[/latex]” or “the common logarithm of [latex]x[/latex].”

  • The logarithm [latex]y[/latex] is the exponent to which [latex]10[/latex] must be raised to get [latex]x[/latex].
  • Since the functions [latex]y=10^{x}[/latex] and [latex]y=\mathrm{log}\left(x\right)[/latex] are inverse functions, [latex]\mathrm{log}\left({10}^{x}\right)=x[/latex] for all [latex]x[/latex] and [latex]10^{\mathrm{log}\left(x\right)}=x[/latex] for [latex]x>0[/latex].
Evaluate [latex]\mathrm{log}(100)[/latex].

To evaluate [latex]\mathrm{log}(100)[/latex], we’re looking for the power to which [latex]10[/latex] must be raised to get [latex]100[/latex]. In other words:

[latex]\begin{align*} \log(100) &= x \\ 10^x &= 100 \\ 100 &= 10^2 \\ 10^x &= 10^2 \\ x &= 2 \\ \log(100) &= 2 \end{align*}[/latex]
Evaluate [latex]\mathrm{log}(110)[/latex].To evaluate [latex]\mathrm{log}(110)[/latex], we need to use a calculator or logarithm tables since [latex]110[/latex] is not a power of [latex]10[/latex].