📘Intermediate Algebra Unit 10 – Exponential & Logarithmic Functions

Key Concepts

• Exponential functions expressed in the form $f(x) = b^x$, where $b$ is a positive constant called the base and $x$ is the exponent or power
• Logarithmic functions defined as the inverse of exponential functions, written as $\log_b(x)$, where $b$ is the base and $x$ is the argument
  • Common logarithms have a base of 10 and are denoted as $\log(x)$
  • Natural logarithms have a base of $e$ (approximately 2.718) and are denoted as $\ln(x)$
• Exponential growth occurs when a quantity increases by a constant factor over equal intervals of time (population growth, compound interest)
• Exponential decay happens when a quantity decreases by a constant factor over equal intervals of time (radioactive decay, medication concentration in the body)
• The domain of an exponential function includes all real numbers, while the range is always positive
• The domain of a logarithmic function is limited to positive real numbers, and the range includes all real numbers

Properties and Rules

• Exponential properties:
  • $b^x \cdot b^y = b^{x+y}$ (product of powers)
  • $\frac{b^x}{b^y} = b^{x-y}$ (quotient of powers)
  • $(b^x)^y = b^{xy}$ (power of a power)
  • $b^0 = 1$ for any non-zero base $b$
  • $b^{-x} = \frac{1}{b^x}$ for any non-zero base $b$
• Logarithmic properties:
  • $\log_b(xy) = \log_b(x) + \log_b(y)$ (sum of logs)
  • $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$ (difference of logs)
  • $\log_b(x^r) = r \cdot \log_b(x)$ (log of a power)
  • $\log_b(1) = 0$ for any base $b$
  • $\log_b(b) = 1$ for any base $b$
• Change of base formula: $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$, where $a$ is any positive base other than 1

Graphing Techniques

• Exponential functions have a horizontal asymptote at $y = 0$ when $0 < b < 1$ and no horizontal asymptote when $b > 1$
  • The graph of $f(x) = b^x$ passes through the point (0, 1) for any base $b$
  • For $b > 1$, the graph increases from left to right, while for $0 < b < 1$, the graph decreases from left to right
• Logarithmic functions have a vertical asymptote at $x = 0$ and no horizontal asymptote
  • The graph of $f(x) = \log_b(x)$ passes through the point (1, 0) for any base $b$
  • For $b > 1$, the graph increases from bottom to top, while for $0 < b < 1$, the graph decreases from top to bottom
• Transformations of exponential and logarithmic functions follow the same rules as other functions (shifts, reflections, stretches, and compressions)
• The graphs of exponential and logarithmic functions with the same base are reflections of each other over the line $y = x$

Solving Equations

• To solve exponential equations with the same base, set the exponents equal to each other and solve for the variable
  • Example: $3^{2x-1} = 3^{4x+3}$ becomes $2x-1 = 4x+3$, which simplifies to $x = -2$
• When exponential equations have different bases, use logarithms to solve
  • Example: $2^x = 5$ can be solved by taking the logarithm of both sides with base 2, giving $x = \log_2(5)$
• To solve logarithmic equations, rewrite the equation in exponential form and then solve for the variable
  • Example: $\log_3(x+2) = 4$ becomes $3^4 = x+2$, which simplifies to $x = 79$
• For equations involving both exponential and logarithmic terms, isolate the exponential or logarithmic term on one side and then apply the appropriate inverse function to both sides

Real-World Applications

• Compound interest: $A = P(1 + \frac{r}{n})^{nt}$, where $A$ is the final amount, $P$ is the principal, $r$ is the annual interest rate, $n$ is the number of times interest is compounded per year, and $t$ is the time in years
• Population growth: $P(t) = P_0e^{kt}$, where $P(t)$ is the population at time $t$, $P_0$ is the initial population, $k$ is the growth rate, and $t$ is the time
• Radioactive decay: $A(t) = A_0e^{-\lambda t}$, where $A(t)$ is the amount of radioactive material at time $t$, $A_0$ is the initial amount, $\lambda$ is the decay constant, and $t$ is the time
• Half-life: the time required for a quantity to reduce to half its initial value, calculated as $t_{1/2} = \frac{\ln(2)}{\lambda}$, where $\lambda$ is the decay constant
• Richter scale: measures the magnitude of earthquakes using logarithms, with each unit increase representing a tenfold increase in amplitude and a 32-fold increase in energy released

Common Mistakes and Tips

• Remember that the base of an exponential or logarithmic function must be positive and not equal to 1
• When solving equations, be careful to apply the appropriate inverse function (exponential or logarithmic) to both sides of the equation
• Pay attention to the domain and range of exponential and logarithmic functions to avoid undefined values or extraneous solutions
  • For example, the argument of a logarithm must be positive, so $\log_2(-3)$ is undefined
• When graphing, keep in mind the key characteristics of exponential and logarithmic functions, such as asymptotes, intercepts, and end behavior
• Double-check your calculations, especially when dealing with negative exponents or logarithms of fractions
• Practice manipulating and simplifying expressions using the properties of exponents and logarithms to build fluency and avoid errors

Practice Problems

1. Simplify the expression: $\log_5(3) + \log_5(x) - \log_5(7)$
2. Solve for $x$: $4^{x-2} = 16$
3. Graph the function $f(x) = 2^{x+1} - 3$ and identify its key features
4. Determine the time it takes for a population of bacteria to triple if it doubles every 4 hours
5. Calculate the magnitude of an earthquake that has an amplitude 100 times greater than an earthquake with a magnitude of 3.5 on the Richter scale
6. Solve for $x$: $\ln(x+3) = 2$
7. A radioactive substance has a half-life of 6 days. If the initial amount is 100 grams, how much will remain after 18 days?

Advanced Topics

• Logarithmic differentiation: a technique used to differentiate functions involving products, quotients, or powers of functions by taking the logarithm of both sides and then differentiating implicitly
• Exponential and logarithmic inequalities: solving inequalities that involve exponential or logarithmic terms by applying the appropriate properties and considering the signs of the expressions
  • For example, since the exponential function is always positive, $e^x > 1$ for all $x > 0$
• Hyperbolic functions: analogous to trigonometric functions but based on the hyperbola instead of the circle, often expressed in terms of exponential functions
  • Hyperbolic sine: $\sinh(x) = \frac{e^x - e^{-x}}{2}$
  • Hyperbolic cosine: $\cosh(x) = \frac{e^x + e^{-x}}{2}$
• Logistic growth model: describes growth that is limited by factors such as resources or competition, using the equation $P(t) = \frac{K}{1 + Ae^{-rt}}$, where $K$ is the carrying capacity, $A$ is a constant related to the initial population, and $r$ is the growth rate
• Gaussian functions: bell-shaped curves that involve exponential terms, often used in probability and statistics, with the general form $f(x) = ae^{-\frac{(x-b)^2}{2c^2}}$, where $a$, $b$, and $c$ are constants that determine the height, center, and width of the curve, respectively