Exponential and Logarithmic Functions

5.4 Logarithmic Functions

Logarithms as Inverses

Logarithms answer the question: what exponent gives this number?

Definition: Logarithm

For $b>0$, $b\ne1$,

$$\log_b(x)=y\iff b^y=x,\quad x>0.$$

Definition: Common and natural logs

  1. Common logarithm: $\log(x)=\log_{10}(x)$
  2. Natural logarithm: $\ln(x)=\log_e(x)$

Example: Convert between exponential and logarithmic form

Rewrite each statement.

  1. $2^5=32$
  2. $\log_3(81)=4$
Show Solution
  1. $2^5=32$ becomes $\log_2(32)=5$.
  2. $\log_3(81)=4$ becomes $3^4=81$.
MyOpenMath: Practice: log and exponential form conversion (placeholder)

Evaluating Logarithms

Use inverse relationships and known powers.

Example: Evaluate logarithms exactly

Evaluate:

  1. $\log_2(16)$
  2. $\log_5\left(\frac{1}{25}\right)$
  3. $\ln(e^3)$
Show Solution
  1. $2^4=16$, so $\log_2(16)=4$.
  2. $\frac{1}{25}=5^{-2}$, so value is $-2$.
  3. $\ln(e^3)=3$.
MyOpenMath: Practice: evaluating logarithms (placeholder)