Question 1

Why can't I take the log of a negative number?

Accepted Answer

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.

Question 2

What's the difference between `log` and `ln` on my calculator?

Accepted Answer

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.

Question 3

How do I remember the laws of logarithms?

Accepted Answer

The laws of logarithms directly mirror the laws of indices. Think about the connection: 
- Multiplication of numbers becomes addition of their powers ($a^m 	imes a^n = a^{m+n}$), 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 $\log(x/y) = \log x - \log y$.

Logarithmic and exponential functions

What you need to know

Logs and Exponentials: Unlocking Growth