Exponential and Logarithmic Functions: Inverses, Properties, and Applications

Inverse Functions

Inverse Relationship and Function Definition

Inverse functions are fundamental in algebra, allowing us to reverse the effect of a function. If a function f is one-to-one, its inverse f-1 exists and is also a function. The inverse relation is obtained by interchanging the domain and range of the original function.

• Inverse Relation: Interchanging the first and second coordinates of each ordered pair in a relation produces the inverse relation.

• Domain and Range: The domain of the original function becomes the range of its inverse, and vice versa.

• Reflection: The graph of the inverse is a reflection of the original function across the line y = x.

Vertical-Line Test and Function Definition

The vertical-line test is used to determine whether a graph represents a function. If a vertical line crosses the graph more than once, the graph does not represent a function.

• Function Definition: Each input (x-value) must correspond to only one output (y-value).

• Non-Function Example: If a graph contains two or more points with the same first coordinate, it is not a function.

Inverse of Quadratic Functions

Quadratic functions, such as y = x2 - 5x, do not always have inverses that are functions. This is because multiple x-values can yield the same y-value, violating the one-to-one requirement.

• One-to-One Functions: A function is one-to-one if different inputs give different outputs.

• Horizontal-Line Test: If a horizontal line crosses the graph more than once, the function is not one-to-one.

• Restricting Domain: Sometimes, restricting the domain allows the inverse to be a function.

Exponential Functions

Definition and Properties

An exponential function is defined as f(x) = ax, where a > 0 and a ≠ 1. The variable is in the exponent, distinguishing it from polynomial functions.

• Continuous and One-to-One: Exponential functions are continuous and one-to-one.

• Domain: $(-\infty, \infty)$

• Range: $(0, \infty)$

• Increasing/Decreasing: Increasing if a > 1, decreasing if 0 < a < 1.

• Horizontal Asymptote: y = 0

• y-intercept: (0, 1)

Exponent Rules

Exponent rules are essential for simplifying expressions involving powers:

• Product Rule: $a^m \cdot a^n = a^{m+n}$

• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

• Power Rule: $(a^m)^n = a^{m \cdot n}$

• Raising a Product to a Power: $(ab)^m = a^m b^m$

• Raising a Quotient to a Power: $\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$

• Zero Exponent: $a^0 = 1$

• Negative Exponent: $a^{-m} = \frac{1}{a^m}$

Applications: Compound Interest

Exponential functions model compound interest, population growth, and decay. The compound interest formula is:

• $A(t) = P\left(1 + \frac{r}{n}\right)^{nt}$

• P: Principal amount

• r: Interest rate (decimal)

• n: Number of compounding periods per year

• t: Time in years

Logarithmic Functions

Definition and Properties

The logarithmic function is the inverse of the exponential function. For y = loga(x), x = ay, where x > 0 and a > 0, a ≠ 1.

• Continuous and One-to-One: Logarithmic functions are continuous and one-to-one.

• Domain: $(0, \infty)$

• Range: $(-\infty, \infty)$

• Vertical Asymptote: x = 0

• x-intercept: (1, 0)

• y-intercept: None

Logarithm Properties

Logarithms have several important properties:

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

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

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

• Change of Base Formula: $\log_b M = \frac{\log_a M}{\log_a b}$

• Other Properties: $\log_a a = 1$, $\log_a 1 = 0$, $a^{\log_a x} = x$, $\log_a a^x = x$

Solving Exponential and Logarithmic Equations

To solve exponential equations, use properties of exponents or convert to logarithmic form. For logarithmic equations, use properties of logarithms or convert to exponential form.

• Base-Exponent Property: $a^x = a^y \iff x = y$

• Change to Logarithmic Form: $a^x = b \implies x = \log_a b$

• Change to Exponential Form: $\log_a x = y \implies x = a^y$

Applications: Growth and Decay Models

Population Growth and Decay

Exponential functions model population growth and decay. The general form is:

• Growth: $P(t) = P_0 e^{kt}$, where $k > 0$

• Decay: $P(t) = P_0 e^{-kt}$, where $k > 0$

• Doubling Time: $T = \frac{\ln 2}{k}$

• Half-Life: $T = \frac{\ln 2}{k}$

Compound Interest (Continuous)

For continuous compounding, the formula is:

• $A(t) = P e^{rt}$

• P: Principal

• r: Interest rate (decimal)

• t: Time in years

Summary Table: Exponential vs. Logarithmic Functions

Property	Exponential Function	Logarithmic Function
Domain	$(-\infty, \infty)$	$(0, \infty)$
Range	$(0, \infty)$	$(-\infty, \infty)$
Asymptote	Horizontal: $y=0$	Vertical: $x=0$
Intercept	y-intercept: (0,1)	x-intercept: (1,0)
Inverse	Logarithmic function	Exponential function

Key Takeaways

• Inverse functions swap domain and range.

• Exponential functions model growth and decay; logarithmic functions are their inverses.

• Properties of exponents and logarithms are essential for solving equations and simplifying expressions.

• Applications include compound interest, population growth, and radioactive decay.