Natural Logarithms
The most frequently used base for logarithms is [latex]e[/latex]. Base [latex]e[/latex] logarithms are important in calculus and some scientific applications; they are called natural logarithms. The base [latex]e[/latex] logarithm, [latex]{\mathrm{log}}_{e}\left(x\right)[/latex], has its own notation, [latex]\mathrm{ln}\left(x\right)[/latex].
Most values of [latex]\mathrm{ln}\left(x\right)[/latex] can be found only using a calculator. The major exception is that, because the logarithm of [latex]1[/latex] is always [latex]0[/latex] in any base, [latex]\mathrm{ln}(1)=0[/latex]. For other natural logarithms, we can use the [latex]\mathrm{ln}[/latex] key that can be found on most scientific calculators. We can also find the natural logarithm of any power of [latex]e[/latex] using the inverse property of logarithms.
natural logarithm
A natural logarithm is a logarithm with base [latex]e[/latex]. We write [latex]{\mathrm{log}}_{e}\left(x\right)[/latex] simply as [latex]\mathrm{ln}\left(x\right)[/latex]. The natural logarithm of a positive number [latex]x[/latex] satisfies the following definition:
For [latex]x>0[/latex],
[latex]y=\mathrm{ln}\left(x\right)\text{ is equivalent to }{e}^{y}=x[/latex]
We read [latex]\mathrm{ln}\left(x\right)[/latex] as, “the logarithm with base [latex]e[/latex] of [latex]x[/latex]” or “the natural logarithm of [latex]x[/latex].”
- The logarithm [latex]y[/latex] is the exponent to which [latex]e[/latex] must be raised to get [latex]x[/latex].
- Since the functions [latex]y=e^{x}[/latex] and [latex]y=\mathrm{ln}\left(x\right)[/latex] are inverse functions, [latex]\mathrm{ln}\left({e}^{x}\right)=x[/latex] for all [latex]x[/latex] and [latex]e^{\mathrm{ln}\left(x\right)}=x[/latex] for [latex]x>0[/latex].