Logarithmic and exponential functions — FAQ

A logarithm, $\log_a x$, asks 'what power must I raise $a$ to, to get $x$?' If the base $a$ is positive, raising it to any real power will always result in a positive number. For example, $2^3=8$, $2^{-3}=1/8$, $2^0=1$. There is no real power you can raise 2 to that will give a negative result. Therefore, the argument of a logarithm must be positive.

Typically, the log button refers to the common logarithm, which has a base of 10 (i.e., $\log_{10}x$). The ln button refers to the natural logarithm, which has a base of $e$ (i.e., $\log_e x$). For solving equations like $a^x=b$, you can use either base, and the change of base formula guarantees you will get the same answer.

The laws of logarithms directly mirror the laws of indices. Think about the connection:

• Multiplication of numbers becomes addition of their powers ($a^m \times a^n = a^{m+n}$), which corresponds to the log rule $\log(xy) = \log x + \log y$.
• Division of numbers becomes subtraction of their powers ($a^m / a^n = a^{m-n}$), which corresponds to $\log(x/y) = \log x - \log y$.