Logarithmic and Exponential Functions: Properties, Applications, and Graphs

Logarithmic Functions

Properties of Exponents and Logarithms

Logarithms are mathematical operations that answer the question: "To what exponent must a base be raised to produce a given number?" They are closely related to exponents and have several useful properties for expanding and simplifying expressions.

• Product Rule: $\log_b(MN) = \log_b M + \log_b N$

• Quotient Rule: $\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N$

• Power Rule: $\log_b(M^p) = p \log_b M$

• Change of Base Formula: $\log_b M = \frac{\log_k M}{\log_k b}$ for any positive base $k \neq 1$

Example: Expand $\log_3 \frac{\sqrt{xz}}{3z}$ using properties:

• $\log_3 \frac{\sqrt{xz}}{3z} = \log_3 \sqrt{xz} - \log_3 3z$

• $= \frac{1}{2} \log_3 x + \frac{1}{2} \log_3 z - (\log_3 3 + \log_3 z)$

Solving Logarithmic Equations

To solve equations involving logarithms, use properties to isolate the variable and, if necessary, exponentiate both sides to remove the logarithm.

• Example: Solve $3 \log_2 x + 1 = 7$

• Subtract 1: $3 \log_2 x = 6$

• Divide by 3: $\log_2 x = 2$

• Exponentiate: $x = 2^2 = 4$

Exponential Functions

Definition and Applications

An exponential function has the form $f(x) = ab^{x}$, where $a$ is the initial value and $b$ is the base (growth/decay factor). Exponential functions model growth and decay in populations, radioactive substances, and finance.

• Growth: $b > 1$

• Decay: $0 < b < 1$

Example: Write the exponential function for a population of 67000 decreasing at 1.67% per year:

• Decay rate: $r = 0.0167$

• Function: $P(t) = 67000(1 - 0.0167)^t$

Example: Increasing at 1.67% per year:

• Growth rate: $r = 0.0167$

• Function: $P(t) = 67000(1 + 0.0167)^t$

Converting Between Logarithmic and Exponential Forms

Logarithmic and exponential forms are interchangeable:

• $\log_b a = x$ is equivalent to $b^x = a$

• Example: $\log_2 8 = x$ means $2^x = 8$

Applications of Logarithms and Exponentials

Earthquake Magnitude Comparison

The Richter scale uses logarithms to compare earthquake magnitudes. The formula $\log\left(\frac{B}{T}\right)$ can be used to determine how many times more powerful one earthquake is than another, assuming $B$ and $T$ are the respective amplitudes.

Population Modeling

Exponential functions are used to model population changes over time. For example, $P = 224e^{0.012t}$ models a population where $t$ is years since 2000.

• To predict the population in 2020: $t = 20$

• Calculate: $P = 224e^{0.012 \times 20}$

pH and Hydrogen Ion Concentration

The pH scale is logarithmic: $\text{pH} = -\log_{10}[H^+]$. To find hydrogen ion concentration:

• $[H^+] = 10^{-\text{pH}}$

• Example: For pH 7.6: $[H^+] = 10^{-7.6}$

• For pH 10.5: $[H^+] = 10^{-10.5}$

Graphing Exponential and Logarithmic Functions

Analysis of Graphs

When graphing functions such as $e^{x-5} - 2$, analyze:

• Domain: All real numbers for exponential functions

• Range: $( -2, \infty )$ for $e^{x-5} - 2$

• Vertical Asymptote (VA): None for exponentials

• Horizontal Asymptote (HA): $y = -2$

• Increasing/Decreasing Intervals: Exponential functions increase if $b > 1$, decrease if $0 < b < 1$

Interpreting Graphs

Given a graph of $h(x) = ab^{x} + c$, identify $a$, $b$, and $c$ by analyzing the y-intercept, asymptote, and growth/decay behavior.

• Exponential Growth: The function rises as $x$ increases ($b > 1$)

• Exponential Decay: The function falls as $x$ increases ($0 < b < 1$)

Modeling with Exponential and Logarithmic Functions

Flu Infection Model

Given $P(t) = \frac{3000}{1 + 299e^{-0.8t}}$, answer questions about initial value, time to reach a certain number, and maximum value.

• Initial Value: $t = 0$

• Maximum Value: As $t \to \infty$, denominator approaches 1, so $P(t) \to 3000$

Summary Table: Key Properties of Logarithms and Exponentials

Property	Logarithms	Exponentials
Form	$\log_b x$	$b^x$
Domain	$x > 0$	All real $x$
Range	All real numbers	$y > 0$
Asymptote	Vertical: $x = 0$	Horizontal: $y = 0$
Growth/Decay	Increases slowly	Rapid increase/decrease