Before you read

Answer first — see what you already know.

What is the domain of $f (x) = lo g_{2} (x + 4)$ ?

A
$x > 0$
B
$x > - 4$
C
$x > 4$
D
All real numbers

Check your answer

Answer:

Before you read

Answer first — see what you already know.

The graph of $y = lo g_{5} (x)$ passes through which point?

A
$(0, 1)$
B
$(1, 0)$
C
$(5, 5)$
D
$(0, 0)$

Check your answer

Answer:

Theory

Logarithmic Functions

The function $y = \log_b(x)$ is the inverse of $y = b^x$ .

y = 2ˣ (exponential) — the inverse of log₂(x)

Key features of $y = \log_b(x)$ (for $b > 1$ ):
- Domain: $x > 0$ (only positive inputs)
- Range: all real numbers
- Passes through $(1, 0)$ always (because $\log_b(1) = 0$ )
- Passes through $(b, 1)$ (because $\log_b(b) = 1$ )
- Vertical asymptote at $x = 0$ (y-axis)
- Increasing (slowly) as $x$ grows

The graph is a reflection of $y = b^x$ across the line $y = x$ .

Example 1

What is the vertical asymptote of $y = \log_3(x - 2)$ ?

The argument must be positive: $x - 2 > 0$ , so $x > 2$

Vertical asymptote at $x = 2$ (shifted right by 2)

x = 2

Tip

ACT Pro Tip

Remember: log graphs always have a vertical asymptote where the argument equals 0, and they always pass through the point where the argument equals 1 (giving $y = 0$ ). For $y = \log_b(x - h) + k$ , the asymptote is $x = h$ and the point $(h+1, k)$ is on the graph.

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