Exponential and Logarithmic Functions: Graphs and Properties

Exponential and Logarithmic Functions

Exponential Functions: Definition and Properties

Exponential functions are mathematical models where the rate of change is proportional to the current value, leading to rapid growth or decay. The general form is:

• Standard form: $f(x) = ab^x$

• Continuous growth form: $f(x) = ae^{rx}$

• Where:

  • a is the initial value (vertical intercept)

  • b is the growth factor ($b = 1 + r$ for percent growth rate $r$)

  • r is the continuous growth rate (as a decimal)

Key Properties:

• The domain is all real numbers: $(-\infty, \infty)$

• The range is $(0, \infty)$ for $a > 0$

• There is a horizontal asymptote at $y = 0$ (unless vertically shifted)

• If $b > 1$, the function shows exponential growth; if $0 < b < 1$, it shows exponential decay

Graphs of Exponential Functions

The graph of an exponential function depends on the values of $a$ and $b$:

• Vertical intercept: $(0, a)$

• Point: $(1, ab)$

• Long run behavior:

  • If $b > 1$, as $x \to \infty$, $f(x) \to \infty$; as $x \to -\infty$, $f(x) \to 0$

  • If $0 < b < 1$, as $x \to \infty$, $f(x) \to 0$; as $x \to -\infty$, $f(x) \to \infty$

Example: $f(x) = 2^x$ (growth), $g(x) = (1/2)^x$ (decay). Notice $g(x)$ is a horizontal reflection of $f(x)$.

Effect of Parameters on the Graph

• Changing $b$ (with $a$ fixed) alters the steepness of the graph. The closer $b$ is to 1, the less steep the graph.

• Changing $a$ (with $b$ fixed) changes the vertical intercept and acts as a vertical stretch/compression.

Transformations of Exponential Functions

Exponential functions can be transformed similarly to other functions:

• Vertical stretch/compression: $af(x)$

• Horizontal shift: $f(x + h)$

• Vertical shift: $f(x) + k$ (shifts the horizontal asymptote to $y = k$)

• Reflection: Negative $a$ reflects across the x-axis; negative exponent reflects across the y-axis

General transformed form: $f(x) = ab^x + c$

The horizontal asymptote is $y = c$.

Exponential Growth and Decay in Applications

• Exponential growth: Population, compound interest, etc.

• Exponential decay: Radioactive decay, depreciation, etc.

Example: If a population grows by 3% per year, $P(t) = P_0(1.03)^t$.

Logarithmic Functions: Definition and Properties

The logarithm is the inverse of the exponential function. For $b > 0$, $b \neq 1$:

• $y = \log_b(x)$ if and only if $b^y = x$

• Common log: $\log_{10}(x)$, written as $\log(x)$

• Natural log: $\log_e(x)$, written as $\ln(x)$

Properties of Logarithms

• Inverse properties:

  • $\log_b(b^x) = x$

  • $b^{\log_b(x)} = x$

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

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

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

• Change of base: $\log_b(A) = \frac{\log_c(A)}{\log_c(b)}$

Solving Exponential and Logarithmic Equations

• To solve $a^x = b$, take the logarithm of both sides and use properties to solve for $x$.

• To solve $\log_b(x) = y$, rewrite as $x = b^y$.

Example: Solve $2^x = 10$:

• Take $\ln$ of both sides: $x \ln 2 = \ln 10$

• $x = \frac{\ln 10}{\ln 2} \approx 3.32$

Graphical Features of Exponential and Logarithmic Functions

• Exponential growth curves rise rapidly for $x > 0$ and approach zero for $x < 0$.

• Exponential decay curves fall rapidly for $x > 0$ and approach zero for large $x$.

• Logarithmic functions increase slowly for large $x$ and are undefined for $x \leq 0$.

Summary Table: Exponential Function Features

Parameter	Effect
$a$	Vertical intercept, vertical stretch/compression
$b$	Growth/decay rate, steepness
$c$	Vertical shift, moves horizontal asymptote

Visual Examples