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Logarithmic Growth — Definition, Formula & Examples

Logarithmic growth is a pattern of increase that never stops but keeps slowing down. Each time you multiply the input by a fixed amount, the output goes up by only a fixed amount.

A function f(x)=log⁡bxf(x) = \log_b x with b>1b > 1 shows logarithmic growth on (0,∞)(0, \infty). It is increasing and unbounded, so lim⁡x→∞log⁡bx=∞\lim_{x \to \infty} \log_b x = \infty. It is also concave down, and its rate of change f′(x)=1xln⁡bf'(x) = \frac{1}{x \ln b} approaches 00. For every p>0p > 0, lim⁡x→∞log⁡bxxp=0\lim_{x \to \infty} \frac{\log_b x}{x^p} = 0. So logarithmic growth is slower than any positive power of xx.

Key Formula

log⁡b(kx)=log⁡bx+log⁡bk\log_b(kx) = \log_b x + \log_b k
Where:
  • bb = the base, with b > 1 for growth
  • xx = the input, x > 0
  • kk = the positive factor the input is multiplied by

Worked Example

Problem: Let f(x)=log⁡2xf(x) = \log_2 x. Find f(8)f(8) and f(16)f(16). Then find the input xx that makes f(x)=20f(x) = 20.
Step 1: Evaluate at 8. Since 23=82^3 = 8:
f(8)=log⁡28=3f(8) = \log_2 8 = 3
Step 2: Double the input to 16. The output rises by only 1, because log⁡22=1\log_2 2 = 1:
f(16)=log⁡216=log⁡28+log⁡22=4f(16) = \log_2 16 = \log_2 8 + \log_2 2 = 4
Step 3: To reach an output of 20, rewrite the logarithmic equation in exponential form:
log⁡2x=20  ⟹  x=220=1,048,576\log_2 x = 20 \implies x = 2^{20} = 1{,}048{,}576
Answer: f(8)=3f(8) = 3 and f(16)=4f(16) = 4. To raise the output from 4 to 20, the input must grow from 16 to 1,048,5761{,}048{,}576.

Why It Matters

In AP Calculus you rank growth rates when you evaluate limits at infinity. A result such as lim⁡x→∞ln⁡xx=0\lim_{x\to\infty} \frac{\ln x}{x} = 0 depends on logs growing slower than polynomials. In computer science, binary search takes about log⁡2n\log_2 n steps. That growth rate is why searching a sorted list of a million items needs only about 20 comparisons.

Common Mistakes

Mistake: Thinking a logarithmic graph levels off at a horizontal asymptote because it flattens out.
Correction: The curve flattens, but it never stops rising. For any number MM, log⁡10x>M\log_{10} x > M whenever x>10Mx > 10^M. Logarithmic functions have a vertical asymptote at x=0x = 0 and no horizontal asymptote.

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