Logarithms

Logarithms are the inverse operation to exponents. They are important to know when instead of multiplying by the same term many times, we are dividing by the same term many times.

Logarithm Definition

We define logarithms as follows:

$\log_b{(x)} = y \Longleftrightarrow b^y = x$

We consider $b$ to be the base of the logarithm.

For example, $\log_2{(8)} = 3$, since $2^3 = 8$

Think first: What is $\log_2{(1)}$?

$\log_2{(1)} = 0$, since $2^0 = 1$

We can also take the logarithm of fractions.

$\log_2{(\frac{1}{8})} = -3$, since $2^{-3} = \frac{1}{8}$

Exponent Rules to Logarithm Rules

Since logarithms are based on exponents, all exponent rules apply. When we multiply numbers, we add the exponents, so

$\log{(m\cdot n)} = \log{(m)} + \log{(n)}$

Likewise, division would be subtraction instead.

Since logarithm is an inverse operation, it can cancel out exponents. When we want to "undo" addition, we subtract. For multiplication, we divide. For exponents, we can use logarithms.

$$16 = 2^x$$

$$\log_2{(16)} = \log_2{(2^x)}$$

$$4 = x$$

Check your understanding: If you had $16 = \log_2{(x)}$, how would you solve for $x$? Be careful that you put things in the right spots. We want $x$ to come out in the end, and we know $y = \log_2{(x)}$.

Think about it: Do you know any other rules with exponents? How would they translate to logarithms?