Chapter 5: Exponential and Logarithmic Functions

Logarithms as inverse exponents, log rules, exponential models, and classic science/finance applications.

Problems Covered

Logarithm TranslationMove between exponential and logarithmic form. Condense the LogProduct, quotient, and power rules. Continuous InterestUse P = Pe^{rt} for continuous growth. Half-Life SampleUse repeated halving for decay. Richter ScaleUse logarithms to compare earthquake intensity.

5.1 What a Logarithm Means

A logarithm asks for an exponent. The expression \log_b(a) means: the exponent on base b that produces a.

\log_b(a)=c \quad \text{means} \quad b^c=a
Core Concept

To translate a logarithm, make the base the base of an exponential equation.

Worked Example 5.1.1 — Logarithm Translation

Rewrite \log_2(32)=5 in exponential form.

  • Identify the base, output, and exponent.b=2,\quad a=32,\quad c=5
  • Use the translation rule.\log_b(a)=c \rightarrow b^c=a
  • Write the exponential statement.2^5=32

Worked Example 5.1.2 — Exponential Translation

Rewrite 3^4=81 in logarithmic form.

  • Identify the base, exponent, and result.b=3,\quad c=4,\quad a=81
  • Use the inverse translation.b^c=a \rightarrow \log_b(a)=c
  • Write the logarithmic statement.\log_3(81)=4

5.2 Logarithm Rules

Log rules come from exponent rules. Multiplication inside a log becomes addition outside, division becomes subtraction, and powers become coefficients.

\log_b(MN)=\log_b(M)+\log_b(N)
\log_b(M/N)=\log_b(M)-\log_b(N)
\log_b(M^p)=p\log_b(M)
Core Concept

To condense logs, move coefficients to exponents first, then combine addition as multiplication.

Worked Example 5.2.1 — Condense the Log

Condense 2\log(x)+\log(3).

  • Move the coefficient into the exponent.2\log(x)=\log(x^2)
  • Use the product rule.\log(x^2)+\log(3)=\log(3x^2)

5.3 Change of Base

Change of base rewrites a logarithm so it can be evaluated with a common calculator key.

\log_b(a)=\frac{\log(a)}{\log(b)}=\frac{\ln(a)}{\ln(b)}
Core Concept

For change of base, put the log of the argument over the log of the base.

Worked Example 5.3.1 — Change of Base

Evaluate \log_5(80).

  • Place the argument in the numerator.\log(80)
  • Place the base in the denominator.\frac{\log(80)}{\log(5)}
  • Evaluate the quotient.\frac{\log(80)}{\log(5)}\approx 2.72

5.4 Classic Exponential and Logarithmic Models

Exponential models appear when a quantity changes by a constant percent or constant factor. Logarithms appear when the unknown is in the exponent or when a scale compares powers.

Core Concept

For continuous growth or decay, substitute into A=Pe^{rt} and solve for the missing quantity.

Worked Example 5.4.1 — Continuous Interest

An account starts with $1200 and earns 5% annual interest compounded continuously for 4 years. Find the account value.

  • Use the continuous growth model.A=Pe^{rt}
  • Substitute principal, rate, and time.A=1200e^{0.05(4)}
  • Simplify the exponent and evaluate.A=1200e^{0.20}\approx 1465.68

The account value is about $1465.68.

5.5 Half-Life Models

A half-life model reduces a quantity by half during each equal time interval.

A=A_0\left(\frac{1}{2}\right)^{t/h}
Core Concept

For half-life, divide elapsed time by the half-life to count how many halvings occur.

Worked Example 5.5.1 — Laboratory Sample

A sample starts at 80 grams and has a half-life of 6 hours. How much remains after 18 hours?

  • Count the number of half-life periods.t/h = 18/6 = 3
  • Apply three halvings.A=80(1/2)^3
  • Evaluate the remaining amount.A=10

5.6 Richter Scale Comparisons

Logarithmic scales compare multiplicative changes. On a base-10 earthquake magnitude scale, a difference of 1 magnitude unit represents a factor of 10 in measured amplitude.

Core Concept

For Richter-style comparisons, subtract magnitudes and raise 10 to that difference.

Worked Example 5.6.1 — Richter Scale Comparison

Compare earthquakes of magnitude 6.4 and 4.4.

  • Subtract the magnitudes.6.4-4.4=2
  • Use base 10 for the comparison factor.10^2=100

The measured amplitude is 100 times as large.