Logarithmic functions are the inverse of exponential functions, allowing us to solve complex equations and model real-world phenomena. They're essential for understanding growth, decay, and scale in fields like physics, chemistry, and finance.

In this section, we'll learn how to convert between logarithmic and exponential forms, evaluate logarithmic expressions, and apply key properties. We'll also explore graphing techniques and practical applications in earthquake measurement, sound intensity, and radioactive decay.

Logarithmic Functions

Logarithmic and exponential conversions

• Logarithmic form $\log_b(x) = y$ equivalent to exponential form $b^y = x$
  • $b$ represents the base, $x$ the argument, and $y$ the logarithm
• Convert from logarithmic to exponential form using $b^{\log_b(x)} = x$
  • Example: $2^{\log_2(8)} = 8$ because $\log_2(8) = 3$ and $2^3 = 8$
• Convert from exponential to logarithmic form using $\log_b(b^x) = x$
  • Example: $\log_5(5^2) = 2$ because $5^2 = 25$ and $\log_5(25) = 2$
• Logarithmic functions are inverse functions of exponential functions

Evaluation of logarithmic expressions

• Common logarithm $\log_{10}(x)$ often written as $\log(x)$
  • $\log(1000) = 3$ because $10^3 = 1000$
  • $\log(0.01) = -2$ because $10^{-2} = 0.01$
• Natural logarithm $\log_e(x)$ often written as $\ln(x)$
  • $e$ mathematical constant approximately 2.71828
  • $\ln(e^4) = 4$ because $e^4 = e^4$
  • $\ln(1) = 0$ because $e^0 = 1$
• Change of base formula $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$ where $a$ any positive base
  • Example: $\log_3(9) = \frac{\ln(9)}{\ln(3)} \approx 2$ because $3^2 = 9$

Properties and applications of logarithms

• Product rule $\log_b(MN) = \log_b(M) + \log_b(N)$
  • $\log_2(8 \cdot 4) = \log_2(8) + \log_2(4) = 3 + 2 = 5$
• Quotient rule $\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$
  • $\log_3(\frac{81}{9}) = \log_3(81) - \log_3(9) = 4 - 2 = 2$
• Power rule $\log_b(M^n) = n \log_b(M)$
  • $\log_2(16^3) = 3 \log_2(16) = 3 \cdot 4 = 12$
• Zero exponent rule $\log_b(1) = 0$
  • $\log_5(1) = 0$ because $5^0 = 1$
• Identity rule $\log_b(b) = 1$
  • $\log_7(7) = 1$ because $7^1 = 7$
• Solving logarithmic equations
  1. Isolate logarithm on one side of equation
  2. Apply exponential function with same base to both sides
  3. Solve resulting exponential equation
     • Example: Solve $\log_2(x) = 3$
       1. $\log_2(x) = 3$
       2. $2^{\log_2(x)} = 2^3$
       3. $x = 8$

Graphing and analysis of logarithms

• Parent function $f(x) = \log_b(x)$ where $b > 0$ and $b \neq 1$
  • Domain $(0, \infty)$, range $(-\infty, \infty)$, vertical asymptote $x = 0$
  • Increasing if $b > 1$, decreasing if $0 < b < 1$
• Transformations
  • Vertical shift $f(x) = \log_b(x) + k$ shifts graph up by $k$ units
  • Horizontal shift $f(x) = \log_b(x - h)$ shifts graph right by $h$ units
  • Vertical stretch/compression $f(x) = a \log_b(x)$
    • Stretches graph vertically by factor of $|a|$ if $|a| > 1$
    • Compresses graph vertically if $0 < |a| < 1$
  • Reflection $f(x) = -\log_b(x)$ reflects graph over $x$-axis

Real-world logarithmic problem solving

• Richter scale $M = \log(\frac{I}{I_0})$
  • $M$ magnitude, $I$ earthquake intensity, $I_0$ reference intensity
  • Example: Earthquake with intensity 1000 times reference has magnitude $\log(1000) = 3$
• Decibel scale $\beta = 10 \log(\frac{I}{I_0})$
  • $\beta$ sound intensity level in decibels, $I$ sound intensity, $I_0$ reference intensity
  • Example: Sound 100 times reference intensity has decibel level $10 \log(100) = 20$ dB
• Exponential growth and decay $A(t) = A_0 e^{kt}$
  • $A(t)$ amount at time $t$, $A_0$ initial amount, $k$ growth or decay constant
  • Half-life $t_{1/2} = \frac{\ln(0.5)}{k}$ where $t_{1/2}$ half-life, $k$ decay constant
  • Example: Radioactive substance half-life 10 days, decay constant $k = \frac{\ln(0.5)}{10} \approx -0.0693$
• pH scale measures acidity using logarithms: $pH = -\log[H^+]$