Exponential and logarithm equivalence

The exponential function and the logarithm function are inverses of each other. This relationship allows us to switch between exponential form and logarithmic form.

Technique: the equivalence relationship

For $a>0$ and $a\ne 1,$ the relationship between exponentials and logarithms is:

$$y=a^x⟺x={\log }_{a}y$$

• $a$ is the base (it remains the base in both forms).
• $x$ is the index or exponent.
• $y$ is the result of the exponential or the argument of the logarithm.

For example, $2^3=8⟺{\log }_{2}8=3.$

In particular

$$y={\mathrm{e}}^x⟺x=\ln y$$

Special logarithms: $\lg$ and $\ln$

Two bases are so common that they have their own shorthand notation:

• Common logarithm: $\lg x={\log }_{10}x$ (base 10)
• Natural logarithm: $\ln x={\log }_{\mathrm{e}}x$ (base $\mathrm{e}\approx 2.718$)

Tip: just as we discussed that $\mathrm{e}$ behaves like any other number, so do logarithmic quantities like $\ln 2$ and $\lg 5.$