Logarithmic and exponential functions — common mistakes

Your calculator's ln button is for the natural logarithm ($\log_e$). The log button is usually for base 10 ($\log_{10}$). For solving equations, it's often best to use ln as it simplifies expressions involving $e$. Remember the key inverse properties: $\ln(e^x) = x$ and $e^{\ln x} = x$ (for $x>0$).

When solving logarithmic equations, always check your final answers by substituting them back into the original equation. This is to ensure that you are not taking the logarithm of a negative number or zero, which is undefined. Any solution that violates this condition must be rejected.

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 $\log(x/y) = \log x - \log y$.