4.5 Logarithmic Properties

Logarithmic properties and applications are key to simplifying complex expressions and solving equations. These tools help us break down tricky problems into manageable pieces, making calculations easier and more intuitive.

Logarithmic equations and functions open up a world of real-world applications. From compound interest to earthquake magnitudes, logs help us model and understand exponential growth and decay in nature, finance, and science.

Logarithmic Properties and Applications

Rules for logarithmic simplification

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• Product rule simplifies the logarithm of a product into the sum of individual logarithms $\log_b(M \cdot N) = \log_b(M) + \log_b(N)$ (e.g., $\log_2(6x) = \log_2(2) + \log_2(3) + \log_2(x)$)
• Quotient rule converts the logarithm of a quotient into the difference of logarithms $\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$ (e.g., $\log_5(\frac{25}{x}) = \log_5(25) - \log_5(x)$)
• Power rule simplifies the logarithm of a number raised to a power by multiplying the logarithm by the exponent $\log_b(M^n) = n \cdot \log_b(M)$ (e.g., $\log_3(x^4) = 4 \log_3(x)$)
• These rules allow for the simplification of complex logarithmic expressions into more manageable forms

Expansion of logarithmic expressions

• Expanding logarithms involves applying the product rule to split the logarithm of a product into a sum of individual logarithms (e.g., $\log_2(8x^3) = \log_2(8) + \log_2(x^3)$)
• The quotient rule separates the logarithm of a quotient into the difference of logarithms (e.g., $\log_b(\frac{x^2}{y}) = \log_b(x^2) - \log_b(y)$)
• Condensing logarithmic expressions combines logarithms with the same base using the sum or difference properties (e.g., $\log_3(x) + \log_3(y) = \log_3(xy)$)
• The power rule simplifies logarithms with exponents by multiplying the logarithm by the power (e.g., $2\log_4(x) = \log_4(x^2)$)

Change-of-base formula for logarithms

• The change-of-base formula $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$ converts a logarithm from one base to another (e.g., $\log_5(x) = \frac{\ln(x)}{\ln(5)}$)
• Commonly used bases include $e$ for the natural logarithm and 10 for the common logarithm
• To evaluate logarithms with different bases, convert the given logarithm to a known base using the change-of-base formula and simplify the resulting expression (e.g., $\log_3(5) = \frac{\log(5)}{\log(3)} \approx 1.465$)
• This formula allows for the evaluation of logarithms without the need for a calculator that supports the specific base

Logarithmic Equations and Functions

Solving logarithmic equations

1. Isolate the logarithmic term on one side of the equation using algebraic techniques while maintaining the equation's balance (e.g., $2\log_3(x) + 1 = 7 \Rightarrow \log_3(x) = 3$)
2. Convert the logarithmic equation to its exponential equivalent using the definition of logarithms $\log_b(x) = y \Leftrightarrow b^y = x$ (e.g., $\log_3(x) = 3 \Rightarrow 3^3 = x$)
3. Solve the resulting exponential equation using appropriate techniques (e.g., $3^3 = x \Rightarrow x = 27$)

• Always check for extraneous solutions by substituting the result back into the original equation

Graphing of logarithmic functions

• The logarithmic function $f(x) = \log_b(x)$ is the inverse function of the exponential function $g(x) = b^x$
• Domain: $(0, \infty)$; Range: $(-\infty, \infty)$
• Vertical asymptote at $x = 0$ since the logarithm is undefined for non-positive numbers
• Increasing if $b > 1$, decreasing if $0 < b < 1$
• Passes through the point $(1, 0)$ since $\log_b(1) = 0$ for any base $b$
• Logarithmic and exponential functions are reflections of each other over the line $y = x$
• Inverse functions undo each other: $\log_b(b^x) = x$ and $b^{\log_b(x)} = x$

Real-world applications of logarithms

• Compound interest formula $A = P(1 + \frac{r}{n})^{nt}$ can be solved for time $t$ using logarithms: $t = \frac{\log(A/P)}{\log(1 + r/n)} \cdot \frac{1}{n}$ (e.g., calculating the time needed to double an investment)
• Exponential growth models for population growth and other natural phenomena can be linearized using logarithms to determine growth rates
• Exponential decay models, such as radioactive decay, use logarithms to calculate half-lives and decay rates
• The Richter scale measures earthquake magnitudes using the logarithmic formula $M = \log(\frac{A}{A_0})$, where $M$ is the magnitude, $A$ is the maximum amplitude of seismic waves, and $A_0$ is a reference amplitude (e.g., an earthquake with a magnitude of 5 has a maximum amplitude 10 times greater than a magnitude 4 earthquake)
• Logarithms are used in various scientific fields, such as chemistry (pH scale), biology (decibel scale), and engineering (signal processing and data compression)