1.5 Exponential and Logarithmic Functions

Exponential Functions

Exponential and logarithmic functions model growth, decay, and many real-world relationships. They're inverses of each other: exponentials grow (or decay) rapidly, while logarithms compress that rapid change into a slower scale. You'll rely on both throughout calculus, so getting comfortable with their shapes, rules, and connections now pays off.

Graphing Exponential Functions

The general form of an exponential function is $f(x) = a \cdot b^x$.

• $a$ controls the vertical stretch and sets the y-intercept at $(0, a)$
• $b$ is the base, and it determines the function's behavior:
  • When $b > 1$ (e.g., 2, $e$, 10), the function grows as $x$ increases
  • When $0 < b < 1$ (e.g., $\frac{1}{2}$, $\frac{1}{e}$), the function decays as $x$ increases

For applied problems, you'll often see these written with a rate:

• Growth: $f(x) = a \cdot (1 + r)^x$, where $r > 0$ is the growth rate (e.g., $r = 0.05$ for 5% growth)
• Decay: $f(x) = a \cdot (1 - r)^x$, where $0 < r < 1$ is the decay rate (e.g., $r = 0.02$ for 2% decay)

Half-life is the time it takes for a decaying quantity to drop to half its initial value. If a substance has a half-life of 10 years and you start with 100 g, you'll have 50 g after 10 years and 25 g after 20 years.

Every exponential function $f(x) = a \cdot b^x$ (with $a > 0$) has a horizontal asymptote at $y = 0$. The graph approaches the x-axis but never touches it.

Comparison of Exponential Bases

All exponential functions of the form $b^x$ pass through the point $(0, 1)$, since any positive base raised to the zero power equals 1.

• For $b > 1$: larger bases produce steeper growth. The graph of $3^x$ climbs faster than $2^x$.
• For $0 < b < 1$: smaller bases produce steeper decay. The graph of $(1/3)^x$ drops faster than $(1/2)^x$.

Significance of the Natural Base $e$

The constant $e \approx 2.71828$ is called the natural base. It shows up naturally whenever growth or decay happens continuously rather than in discrete steps.

The function $f(x) = e^x$ is the natural exponential function. It's special because its rate of change at any point equals its value at that point, a property you'll use heavily in derivatives.

Common applications of $e^x$:

• Continuous compound interest: $A = Pe^{rt}$, where $P$ is principal, $r$ is the annual rate, and $t$ is time in years
• Population growth: modeling bacteria or viral spread where reproduction is continuous
• Radioactive decay: carbon dating and nuclear physics use $e^{-kt}$ to model how substances break down over time

Logarithmic Functions

Logarithmic Functions and Graphs

A logarithmic function has the form $f(x) = \log_b(x)$, where $b > 0$ and $b \neq 1$. The logarithm $\log_b(x)$ answers the question: "What exponent do you put on $b$ to get $x$?"

For example, $\log_2(8) = 3$ because $2^3 = 8$.

Two bases come up constantly:

• Common logarithm (base 10): written as $\log(x)$
• Natural logarithm (base $e$): written as $\ln(x)$

Key features of the graph of $\log_b(x)$:

• Domain: $x > 0$ only (you can't take the log of zero or a negative number)
• Range: all real numbers
• Vertical asymptote at $x = 0$ (the graph approaches the y-axis but never crosses it)
• Passes through $(1, 0)$, since $\log_b(1) = 0$ for any valid base

Exponential vs. Logarithmic Functions

Exponentials and logarithms are inverses of each other. This means they undo each other:

$b^{\log_b(x)} = x \quad \text{and} \quad \log_b(b^x) = x$

The inverse relationship also means: if $y = b^x$, then $x = \log_b(y)$. For instance, if $y = 2^x$, then $x = \log_2(y)$.

Graphically, the curves of $y = b^x$ and $y = \log_b(x)$ are mirror images of each other across the line $y = x$. This reflection is true for any pair of inverse functions.

Change of Base Formula

Most calculators only have buttons for $\log$ (base 10) and $\ln$ (base $e$). To evaluate a logarithm with any other base, use the change of base formula:

$\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$

where $a$ is any convenient base ($a > 0$, $a \neq 1$).

For example, to find $\log_5(20)$ on a calculator:

$\log_5(20) = \frac{\ln(20)}{\ln(5)} \approx \frac{3.0}{1.609} \approx 1.861$

You can use either $\ln$ or $\log$ in the formula; just be consistent in the numerator and denominator.

Hyperbolic Functions

Hyperbolic functions are built from combinations of $e^x$ and $e^{-x}$. They're analogous to trig functions but are defined using exponentials instead of circles. You'll encounter them later in calculus when dealing with certain integrals and differential equations.

• Hyperbolic sine: $\sinh(x) = \frac{e^x - e^{-x}}{2}$
  • Odd function (symmetric about the origin), domain and range are all real numbers
• Hyperbolic cosine: $\cosh(x) = \frac{e^x + e^{-x}}{2}$
  • Even function (symmetric about the y-axis), domain is all real numbers, range is $y \geq 1$
• Hyperbolic tangent: $\tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$
  • Odd function, domain is all real numbers, range is $-1 < y < 1$
  • Has horizontal asymptotes at $y = 1$ and $y = -1$

A useful identity that mirrors the Pythagorean identity from trig: $\cosh^2(x) - \sinh^2(x) = 1$.

Exponent and Logarithm Rules

Exponent Rules

These rules apply whenever the base $a > 0$:

• Product rule: $a^m \cdot a^n = a^{m+n}$
• Quotient rule: $\frac{a^m}{a^n} = a^{m-n}$
• Power rule: $(a^m)^n = a^{mn}$
• Zero exponent: $a^0 = 1$ (for $a \neq 0$)
• Negative exponent: $a^{-n} = \frac{1}{a^n}$

Logarithm Rules

These rules apply for $b > 0$, $b \neq 1$, and positive arguments $x, y$:

• Product rule: $\log_b(xy) = \log_b(x) + \log_b(y)$
• Quotient rule: $\log_b\!\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$
• Power rule: $\log_b(x^n) = n\log_b(x)$
• Change of base: $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$

Notice the pattern: multiplication inside a log becomes addition outside, division becomes subtraction, and exponents become coefficients. These rules are just the exponent rules translated into log language.

Solving Exponential and Logarithmic Equations

Exponential equations (variable in the exponent) are solved by taking a logarithm of both sides:

1. Isolate the exponential expression
2. Apply $\ln$ or $\log$ to both sides
3. Use the power rule to bring the variable down
4. Solve for the variable

Example: Solve $3^x = 20$. Take $\ln$ of both sides: $x \ln(3) = \ln(20)$, so $x = \frac{\ln(20)}{\ln(3)} \approx 2.727$.

Logarithmic equations (variable inside a log) are solved by converting to exponential form:

1. Use log rules to combine into a single logarithm if needed
2. Rewrite in exponential form: $\log_b(x) = c$ becomes $x = b^c$
3. Solve for the variable
4. Check your answer, since log arguments must be positive (reject any solution where the argument would be zero or negative)

Example: Solve $\log_2(x - 1) = 4$.

Rewrite as $x - 1 = 2^4 = 16$, so $x = 17$. Check: $17 - 1 = 16 > 0$, so the solution is valid.