対数と指数関数

A logarithm is just an exponential statement rewritten the other way round. For a base $a$ (with $a>0,a\ne 1$), ${\log }_{a}x$ answers the question 'to what power must I raise $a$ to get $x$?'. The whole definition is captured by the equivalence below. Note that the argument $x$ must always be positive.

Re-expressing the index laws such as $a^m{a}^n=a^{m+n}$ gives the three laws of logs below. They only apply when the bases match. The key idea: a product becomes a sum, a quotient becomes a difference, and a power becomes a coefficient. You should also know ${\log }_{a}a=1$ and ${\log }_{a}1=0$.

A logarithm to base $e=2.71828\dots$ is called a natural logarithm, and ${\log }_{e}x$ is written $\ln x$. The number $e$ is the base of the function $e^x$, which is unchanged by differentiation, and it appears naturally in growth and decay models. Because $\ln$ and $e^x$ are inverse functions, $\ln e^x=x$ and $e^{\ln x}=x$. These two identities do the heavy lifting when solving exponential equations.

When the unknown sits in the exponent, as in $a^x=b$, taking logs of both sides brings the exponent down to the front. Any base works, but since calculators carry $\ln$ (and the common log $\log$), using $\ln$ is the practical choice. From $x\ln a=\ln b$ you obtain $x=\frac{\ln b}{\ln a}$.

When experimental data is believed to follow $y=A{x}^n$ or $y=A{b}^x$, taking logs turns the relationship into a straight line, and the constants come from the gradient and intercept. For $y=A{x}^n$, $\ln y=\ln A+n\ln x$, so plotting $\ln y$ against $\ln x$ gives a line of gradient $n$ and intercept $\ln A$. For $y=A{b}^x$, $\ln y=\ln A+x\ln b$, so plotting $\ln y$ against $x$ gives a line of gradient $\ln b$ and intercept $\ln A$.