Logarithmic functions are powerful tools that help us understand exponential relationships. They're the flip side of exponents, letting us solve complex problems in fields like science and finance.

These functions have unique properties that make them incredibly useful. From measuring earthquakes to calculating compound interest, logarithms simplify complex calculations and help us make sense of vast ranges of data.

Logarithmic Functions

Logarithmic and exponential form conversion

• Logarithmic form $\log_b(x) = y$ equivalent to exponential form $b^y = x$
  • $b$ represents the base, $x$ the argument, and $y$ the exponent or logarithm
• Convert logarithmic to exponential form by using the base as the exponent's base, the logarithm as the exponent, and the argument as the result ($2^3 = 8$)
• Convert exponential to logarithmic form by using the exponent's base as the logarithm's base, the result as the argument, and the exponent as the logarithm ($\log_2(8) = 3$)
  • This conversion demonstrates the inverse function relationship between logarithmic and exponential functions

Evaluation of logarithms with bases

• Change of base formula $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$, where $a$ represents any base
  • Evaluates logarithms with any base using a calculator with a different base (usually base 10 or $e$)
• Common bases include base 10 (common logarithm) $\log(x)$ or $\log_{10}(x)$ and base $e$ (natural logarithm) $\ln(x)$ or $\log_e(x)$, where $e \approx 2.71828$
• Logarithms can only be evaluated for positive arguments ($x > 0$)

Domain, Range, and Asymptotes of Logarithmic Functions

• Domain of logarithmic functions: all positive real numbers (x > 0)
• Range of logarithmic functions: all real numbers
• Vertical asymptote: x = 0, as the function approaches negative infinity as x approaches 0 from the right

Applications of Logarithms

Common logarithms in real-world applications

• Richter scale measures the magnitude of an earthquake
  • Richter magnitude calculated by $\log(\frac{\text{Seismic wave amplitude}}{\text{Standard amplitude}})$
  • An increase of 1 on the Richter scale corresponds to a tenfold increase in seismic wave amplitude
• Decibel scale measures the intensity of sound
  • Decibels (dB) calculated by $10 \log(\frac{\text{Sound intensity}}{\text{Reference intensity}})$
  • An increase of 10 dB corresponds to a tenfold increase in sound intensity
• pH scale measures the acidity or alkalinity of a solution
  • pH calculated by $-\log([\text{H}^+])$, where $[\text{H}^+]$ represents the concentration of hydrogen ions in moles per liter
  • A decrease of 1 pH unit corresponds to a tenfold increase in hydrogen ion concentration
• These applications demonstrate the use of logarithmic scales in various scientific fields

Natural logarithms for growth and decay

• Exponential growth modeled by $A(t) = A_0e^{kt}$, where $A_0$ represents the initial amount, $k$ the growth rate, and $t$ time
  • Doubling time calculated by $t_d = \frac{\ln(2)}{k}$
• Exponential decay modeled by $A(t) = A_0e^{-kt}$, where $A_0$ represents the initial amount, $k$ the decay rate, and $t$ time
  • Half-life calculated by $t_{1/2} = \frac{\ln(2)}{k}$
• Carbon dating determines the age of organic materials based on the decay of carbon-14
  • Remaining carbon-14 calculated by $A(t) = A_0e^{-kt}$, where $k = \frac{\ln(2)}{5730}$ (half-life of carbon-14 is 5730 years)

Solving equations with logarithmic properties

• Product rule $\log_b(MN) = \log_b(M) + \log_b(N)$
• Quotient rule $\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$
• Power rule $\log_b(M^n) = n\log_b(M)$
• Steps to solve logarithmic equations:
  1. Isolate the logarithm on one side of the equation
  2. Apply the appropriate logarithmic properties to simplify the equation
  3. Convert the equation to exponential form
  4. Solve the resulting exponential equation