Logarithms Quick Review

Logarithms: Definition and Inverse Relationship

• The logarithm (base $b$) of $x$ is the exponent $y$ to which $b$ must be raised to get $x$:
  $\log_b x = y \iff b^y = x$
• Logarithms are the inverse of exponential functions with the same base.
• Notation:
  • Common log (base 10): $\log x$ (i.e., $\log_{10} x$)
  • Natural log (base $e$): $\ln x$ (i.e., $\log_e x$)

Properties of Logs

• Inverse properties:
  • $\log_b (b^x) = x$
  • $b^{\log_b x} = x$
• Exponent (power) property:
  • $\log<em>b (a^r) = r \log</em>b a$
• Change of base (calculator-friendly):
  • $\log<em>b x = \frac{\ln x}{\ln b} = \frac{\log</em>{10} x}{\log_{10} b}$

Converting Between Exponential and Logarithmic Form

• If $b^c = a$, then $\log_b a = c$.
• If $\log_b a = c$, then $b^c = a$.

Evaluating and Solving with Logs

• Steps to solve exponential equations:
  1. Isolate the exponential expression.
  2. Take the logarithm of both sides.
  3. Use log properties to pull the exponent out.
  4. Solve for the unknown variable.
• Example: Evaluating common logs:
  • $\log(1000) = 3$ since $10^3 = 1000$
• Example: Solving an exponential equation:
  • Solve $5 e^{-0.3 t} = 2$:
    $e^{-0.3 t} = \frac{2}{5}$
    $-0.3 t = \ln\left(\frac{2}{5}\right)$
    $t = -\frac{1}{0.3}\ln\left(\frac{2}{5}\right) \approx 3.054$
• Real-world application: pH:
  • $\text{pH} = -\log([H^+])$
  • If $[H^+] = 10^{-4}$ (e.g., vinegar), then $\text{pH} = -\log(10^{-4}) = 4$. (Lower pH = higher acidity)

Graphical Features of the Logarithm ($g(x) = \log_b x$)

• Domain: $x > 0$
• Horizontal intercept: $(1,0)$
• Vertical asymptote: $x = 0$
• Range: $(-\infty, \infty)$
• Shape: increasing and concave down for base $b>1$
• Passes through: $(1,0)$ and $(b,1)$

Domain Considerations

• Input to the log must always be positive.
• Example: For $f(x) = \log(5 - 2x)$, require $5 - 2x > 0 \implies x < 2.5$.

Quick Reference

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