Study Guide: Inverse, Exponential, and Logarithmic Functions (Precalculus)

Ch. 4 – Inverse, Exponential, and Logarithmic Functions

Introduction to Exponential Functions

Exponential functions are a fundamental class of functions in mathematics, characterized by a variable exponent. They are essential for modeling growth and decay in various fields.

• Definition: An exponential function has the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

• Polynomial vs. Exponential: In polynomials, the variable is the base; in exponentials, the variable is the exponent.

• Example: $f(x) = 2^x$ is exponential; $f(x) = x^2$ is polynomial.

The Number e

The number e is a mathematical constant approximately equal to 2.71828. It is the base of the natural exponential function and arises in many contexts involving continuous growth.

• Definition: $e \approx 2.71828$

• Exponential Function with Base e: $f(x) = e^x$

• Applications: Compound interest, population growth, and many natural processes.

• Example: Evaluate $e^2$ using a calculator.

Graphing Exponential Functions

Exponential functions have distinctive graphs, which depend on the base. They exhibit rapid growth or decay and have important properties regarding domain, range, and asymptotes.

• General Form: $f(x) = a^x$

• Domain: $(-\infty, \infty)$

• Range: $(0, \infty)$

• Horizontal Asymptote: $y = 0$

• Growth vs. Decay:

  • If $a > 1$, the function increases (growth).

  • If $0 < a < 1$, the function decreases (decay).

• Example: Sketch $f(x) = 2^x$ and $f(x) = (1/2)^x$.

Transformations of Exponential Graphs

Exponential graphs can be transformed using shifts, reflections, and stretches/compressions. These transformations follow the same principles as those for other functions.

• General Transformation: $g(x) = a^{x-h} + k$

• Horizontal Shift: $h$ units right if $h > 0$, left if $h < 0$

• Vertical Shift: $k$ units up if $k > 0$, down if $k < 0$

• Reflection: Over the $x$-axis: $-a^x$; over the $y$-axis: $a^{-x}$

• Example: Graph $g(x) = 2^{x-3} + 1$

Introduction to Logarithms

Logarithms are the inverses of exponential functions. They answer the question: "To what exponent must the base be raised to produce a given number?"

• Definition: $y = \log_a x$ means $a^y = x$

• Common Logarithms: Base 10, written as $\log x$

• Natural Logarithms: Base $e$, written as $\ln x$

• Example: $\log_2 8 = 3$ because $2^3 = 8$

Properties of Logarithms

Logarithms obey several important properties that simplify expressions and solve equations.

Name	Property	Description
Product Rule	$\log_a (xy) = \log_a x + \log_a y$	Log of a product equals sum of logs
Quotient Rule	$\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$	Log of a quotient equals difference of logs
Power Rule	$\log_a (x^r) = r \log_a x$	Log of a power equals exponent times log

• Example: $\log_2 8 = \log_2 (2^3) = 3 \log_2 2 = 3$

Change of Base Property

The change of base formula allows you to evaluate logarithms with any base using common or natural logarithms.

• Formula: $\log_a x = \frac{\log_b x}{\log_b a}$ (where $b$ is any positive base, commonly 10 or $e$)

• Example: $\log_2 7 = \frac{\log 7}{\log 2}$

Graphing Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. Their graphs have distinct features, including vertical asymptotes and specific domains and ranges.

• General Form: $f(x) = \log_a x$

• Domain: $(0, \infty)$

• Range: $(-\infty, \infty)$

• Vertical Asymptote: $x = 0$

• Example: Graph $f(x) = \log_2 x$

Transformations of Logarithmic Graphs

Logarithmic graphs can be shifted, reflected, and stretched/compressed similarly to exponential graphs.

• General Transformation: $g(x) = \log_a (x-h) + k$

• Horizontal Shift: $h$ units right if $h > 0$, left if $h < 0$

• Vertical Shift: $k$ units up if $k > 0$, down if $k < 0$

• Reflection: Over the $x$-axis: $-\log_a x$; over the $y$-axis: $\log_a (-x)$

• Example: Graph $g(x) = \log_2 (x-3) + 1$

Solving Exponential and Logarithmic Equations

Equations involving exponentials and logarithms can be solved using properties of exponents and logarithms, and sometimes by rewriting both sides with the same base.

• Solving Exponential Equations:

  • If possible, rewrite both sides with the same base and set exponents equal.

  • Example: $2^x = 8 \Rightarrow 2^x = 2^3 \Rightarrow x = 3$

  • If not possible, use logarithms: $a^x = b \Rightarrow x = \log_a b$

• Solving Logarithmic Equations:

  • Rewrite in exponential form: $\log_a x = b \Rightarrow x = a^b$

  • Example: $\log_3 x = 2 \Rightarrow x = 3^2 = 9$

Common Powers Table

This table lists common squares, cubes, and other powers for reference when solving equations.

Squares	Cubes	Other
4 = $2^2$ 9 = $3^2$ 16 = $4^2$ 25 = $5^2$ 36 = $6^2$ 49 = $7^2$ 64 = $8^2$ 81 = $9^2$ 100 = $10^2$	8 = $2^3$ 27 = $3^3$ 64 = $4^3$ 125 = $5^3$ 216 = $6^3$ 343 = $7^3$ 512 = $8^3$ 729 = $9^3$ 1000 = $10^3$	32 = $2^5$ 81 = $3^4$ 243 = $3^5$ 256 = $4^4$ 625 = $5^4$ 1024 = $2^{10}$ 4096 = $2^{12}$ 3125 = $5^5$

Summary

• Exponential and logarithmic functions are inverses.

• Graphing and transforming these functions is essential for understanding their behavior.

• Properties of logarithms simplify expressions and solve equations.

• Solving equations may require rewriting, applying properties, or using the change of base formula.

Additional info: This guide expands on brief notes and examples to provide a comprehensive overview suitable for Precalculus students.