Exponential and Logarithmic Functions: Graphs, Properties, and Applications

Exponential and Logarithmic Functions

Graphs and Transformations of Exponential Functions

Exponential functions have the general form $y = a imes b^{x} + k$, where $a$ is a vertical stretch/compression, $b$ is the base, and $k$ is a vertical shift. Understanding their graphs and transformations is essential for analyzing their behavior.

• Y-intercept: The point where the graph crosses the y-axis (set $x=0$).

• Domain: The set of all possible $x$-values (usually $(-\infty, \infty)$ for basic exponentials).

• Range: The set of all possible $y$-values (often $(0, \infty)$ or shifted by $k$).

• Horizontal Asymptote: The value $y$ approaches as $x \to \infty$ or $x \to -\infty$ (typically $y = k$).

• Transformations: Shifts, reflections, and stretches/compressions from the parent function $y = b^x$.

Example: For $y = 3^{x-2} - 2$:

• Shift right 2 units, down 2 units.

• Domain: $(-\infty, \infty)$

• Range: $(-2, \infty)$

• Horizontal asymptote: $y = -2$

Exponential Growth and Decay Applications

Exponential functions model real-world phenomena such as population growth, radioactive decay, and compound interest. The general form is $y = y_0 e^{kt}$ or $y = y_0 (1 + r)^t$.

• Growth: $k > 0$ or $r > 0$

• Decay: $k < 0$ or $r < 0$

• Applications: Savings accounts, atmospheric pressure, radioactive isotopes, population models.

Example: If $y = 1172.2e^{0.07t}$, then $y_0 = 1172.2$, $k = 0.07$ (growth rate).

Logarithmic Functions: Properties and Graphs

Logarithmic functions are the inverses of exponential functions. The general form is $y = \log_b(x)$, where $b$ is the base.

• Domain: $x > 0$ (for $\log_b(x)$)

• Range: $(-\infty, \infty)$

• Vertical Asymptote: $x = 0$ (or shifted by $h$ in $y = \log_b(x-h)$)

• Transformations: Shifts, reflections, and stretches/compressions from the parent function $y = \log_b(x)$

Example: For $y = \log_2(x-5) - 1$:

• Shift right 5 units, down 1 unit.

• Domain: $x > 5$

• Range: $(-\infty, \infty)$

• Vertical asymptote: $x = 5$

Evaluating and Manipulating Logarithms

Logarithms can be evaluated, expanded, or condensed using their properties:

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

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

• Power Rule: $\log_b(M^k) = k\log_b(M)$

• Change of Base Formula: $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$

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

Expanding and Condensing Logarithmic Expressions

Logarithmic expressions can be rewritten using the properties above to either expand (write as a sum/difference of logs) or condense (combine into a single log).

• Expand: $\log_b\left(\frac{uv^3}{w^2}\right) = \log_b(u) + 3\log_b(v) - 2\log_b(w)$

• Condense: $2\log_b(x) + 3\log_b(y) = \log_b(x^2y^3)$

Rewriting Equations: Exponential and Logarithmic Forms

Equations can be rewritten between exponential and logarithmic forms:

• Exponential to Logarithmic: $b^y = x \implies y = \log_b(x)$

• Logarithmic to Exponential: $y = \log_b(x) \implies b^y = x$

Example: $5^2 = 25 \implies 2 = \log_5(25)$

Table: Properties of Exponential and Logarithmic Functions

Function	Domain	Range	Asymptote	Intercept
$y = b^x$	$(-\infty, \infty)$	$(0, \infty)$	Horizontal: $y=0$	$(0,1)$
$y = \log_b(x)$	$(0, \infty)$	$(-\infty, \infty)$	Vertical: $x=0$	$(1,0)$

Additional info:

• Some problems involve sketching graphs and identifying transformations, which is a key skill for understanding function behavior.

• Applications include both continuous and discrete exponential models, as well as logarithmic equations for solving for time or rate in growth/decay problems.