Mathfolis

Logarithmic Functions

Unit 2 · Exponential and Logarithmic Functions

What AP Precalc asks here

A logarithmic function f(x)=log⁡bxf(x) = \log_b x has domain (0,∞)(0, \infty), range R\mathbb{R}, vertical asymptote (VA) x=0x = 0, and x-intercept (1,0)(1, 0). When b>1b > 1, the function is increasing and concave down; when 0<b<10 < b < 1, it is decreasing. Logs grow extremely slowly — slower than any positive power of x. Transformations shift and stretch the graph the same way as for any other function.

Base form f(x)=log⁡bxf(x) = \log_b x with b>1b > 1

Domain
(0,∞)(0, \infty)
Range
R\mathbb{R}
Vertical asymptote
x=0x = 0
X-intercept
(1,0)(1, 0)

Transformations

Horizontal shift
log⁡b(x−h)  ⇒  VA at x=h\log_b(x - h) \;\Rightarrow\; \text{VA at } x = h
Vertical shift
log⁡bx+k  ⇒  shifted up by k\log_b x + k \;\Rightarrow\; \text{shifted up by } k
AP Tip: Log growth eventually loses to every polynomial: log⁡bx=o(xp)\log_b x = o(x^p) for any p>0p > 0 as x→∞x \to \infty. That's why log scales compress wide data ranges.
Type 1

Read key features

Apply the horizontal shift to the asymptote, the vertical shift to the x-intercept, and the domain constraint to the argument.

Example 1
For f(x)=log⁡2(x+3)f(x) = \log_2(x + 3), state the domain, vertical asymptote, and x-intercept. (A) Domain R\mathbb{R}, VA x=3x = 3, x-intercept (3,0)(3, 0) (B) Domain (−3,∞)(-3, \infty), VA x=−3x = -3, x-intercept (−2,0)(-2, 0) (C) Domain (3,∞)(3, \infty), VA x=3x = 3, x-intercept (4,0)(4, 0) (D) Domain (0,∞)(0, \infty), VA x=0x = 0, x-intercept (1,0)(1, 0)

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Type 2

Apply transformations of log functions

Solve for the x-intercept by setting log⁡b(stuff)=0\log_b(\text{stuff}) = 0, i.e., the argument equals 1.

Example 2
The graph of g(x)=log⁡3x−2g(x) = \log_3 x - 2 crosses the x-axis at what x-value? (A) x=1x = 1 (B) x=3x = 3 (C) x=9x = 9 (D) x=8x = 8

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Type 3

Compare growth behavior and concavity

For b>1b > 1, log growth is slow and concave down. For 0<b<10 < b < 1, the log is decreasing.

Example 3
Is f(x)=log⁡2xf(x) = \log_2 x concave up or concave down? (A) Concave up (B) Concave down (C) Linear (D) Switches between concavities

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Logarithmic Functions | AP Precalculus — Mathfolis