Exponents and Logarithms

This is an area where drilling can make a huge difference. That’s for two reasons:

1. Contestants often make “small” mistakes here that cost them entire problems
2. Even when getting these problems right, they routinely take far longer than they should

The big question

Are you facile enough with exponents and logarithms to do great on the AMC?

In order to answer “yes,” you should know or have all of the following, without any hints or prompting:

• the basic rules of logarithms
• the basic rules of exponents
• the ability to convert exponential equations into logarithmic form and vice-versa

This means that you should be able to algebraically combine and manipulate exponents and logarithms without having to stop and think for very long in the middle of the process. In addition, you should be able to see at least two or three different possible approaches for any problem involving logarithms or exponents in a significant way.

The logarithm rules

For logarithms, they are as follows:

• ${\log }_{b}c=ANSWER⟷b^{ANSWER}=c$

• $\log x+\log y=\log xy$

• $\log x-\log y=\log \frac{x}{y}$

• $\log a^b=\log (a^b)=b\log a$

• ${\log }_{b}a=\frac{\log a}{\log b}$

• $\ln x={\log }_{e}x$

• $\log a=\log b⟷a=b$

The exponent rules

For exponents, they are as follows:

• $x^a{x}^b=x^{a+b}$

• $\frac{x^a}{x^b}=x^{a-b}$

• $(xy{)}^a=x^a{y}^a$

• $(x^a{)}^b=x^{ab}=(x^b{)}^a$

• $(\frac{x}{y}{)}^a=\frac{x^a}{y^a}$

• $x^{-n}=\frac{1}{x^n}$

• $x^{\frac{m}{n}}=\sqrt[n]{x^m}=(\sqrt[n]{x}{)}^m$