Logarithmic Functions

Video lesson · from 9:112 worked examples

One lesson video covers all of Functions & Graphs, so it opens at 9:11 for this topic — not from the beginning.

A logarithmic function is a function of the form $f(x) = \log_a x$ where $a > 0$ and $x > 0$ .

The relationship between exponential functions and logarithmic functions can be expressed as:

y = a^x \iff \log_a y = x

A logarithm is an index in disguise: $\log_a x = y$ means exactly $a^y = x$ . Being able to switch between these two forms instantly is what most log questions actually test.
You cannot take the log of zero or a negative: so check that any solution keeps every argument positive, and discard those that do not.
$\log_a 1 = 0$ and $\log_a a = 1$ : these two special values shortcut a great many calculations.
Log and exponential are inverses: which is why their graphs are reflections in $y = x$ .

Example 1

Evaluating Logarithmic Expressions

a) Write $5^3 = 125$ in logarithmic form.

b) Evaluate $\log_4 16$ .

a) Using $a^x = y \implies \log_a y = x$ :

\log_5 125 = 3

b) Let $\log_4 16 = x$ .

Rearrange into exponential form: $4^x = 16$ .

Since $4^2 = 16$ , $x = 2$ .

\log_4 16 = 2

Example 2

Basic Logarithmic Equations

a) Solve $\log_5 x = 3$ .

b) Solve $\log_x 81 = 2$ .

a) Convert to exponential form:

x = 5^3 \implies x = 125

b) Convert to exponential form:

x^2 = 81

x = 9

(Note: base $x$ must be > 0, so ignore -9)