Chapter 4: Exponential and Logarithmic Functions

Vocabulary practice flashcards covering exponential and logarithmic definitions, graphs, properties, formulas, and solving techniques from Chapter 4.

Exponential function

A function of the form $f(x) = a b^x$, where $a$ is a non-zero real number called the initial value, $b$ is any positive real number such that $b \neq 1$, the domain is all real numbers, and the horizontal asymptote is $y = 0$.

Initial value ($a$)

The non-zero real number constant $a$ in $f(x) = a b^x$ that specifies the $y$-intercept of the exponential function at $(0, a)$.

Compound Interest formula

The formula $A(t) = P\left(1 + \frac{r}{n}\right)^{nt}$, which computes the total amount accrued after time $t$ when principal $P$ is invested at an annual interest rate $r$ compounded $n$ times per year.

Continuous Growth/Decay formula

The formula $A(t) = a e^{rt}$, representing continuous growth (if $r > 0$) or decay (if $r < 0$), where $a$ is the initial amount and $r$ is the continuous rate of growth or decay.

Parent exponential function ($f(x) = b^x$)

A one-to-one function where $b > 0$ and $b \neq 1$, characterized by domain $(-\infty, \infty)$, range $(0, \infty)$, horizontal asymptote $y = 0$, $y$-intercept $(0, 1)$, no $x$-intercept, increasing when $b > 1$, and decreasing when $0 < b < 1$.

Logarithmic function

For positive numbers $x > 0$, $b > 0$, and $b \neq 1$, the function defined by $y = \log_b(x)$, which is equivalent to $b^y = x$, meaning the logarithm $y$ is the exponent to which base $b$ must be raised to produce $x$.

Parent logarithmic function ($f(x) = \log_b(x)$)

A one-to-one function characterized by vertical asymptote $x = 0$, domain $(0, \infty)$, range $(-\infty, \infty)$, $x$-intercept $(1, 0)$, key point $(b, 1)$, and no $y$-intercept.

Product rule for logarithms

The logarithmic property stating that $\log_b(MN) = \log_b(M) + \log_b(N)$ for $b > 0$, allowing the logarithm of a product to be rewritten as a sum of individual logarithms.

Quotient rule for logarithms

The logarithmic property stating that $\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)$, allowing the logarithm of a quotient to be rewritten as the difference of individual logarithms.

Power rule for logarithms

The property of logarithms used to rewrite the exponent of an argument as a coefficient multiplying the logarithm, or conversely to condense a multiple of a logarithm into an exponent.

Change-of-base formula

A formula stating that for positive real numbers $M$, $b$, and $n$ (with $n \neq 1$ and $b \neq 1$), $\log_b(M) = \frac{\log_n(M)}{\log_n(b)}$, commonly calculated as $\frac{\ln(M)}{\ln(b)}$ or $\frac{\log(M)}{\log(b)}$.

One-to-one property of exponential functions

The property stating that for any algebraic expressions $S$ and $T$, and any positive real number $b \neq 1$, $b^S = b^T$ if and only if $S = T$.

One-to-one property of logarithms

The property stating that for any algebraic expressions $S$ and $T$, and any positive real number $b \neq 1$, $\log_b(S) = \log_b(T)$ if and only if $S = T$.

Extraneous solution

A solution derived algebraically when solving an equation (such as a logarithmic equation) that does not satisfy the original equation or is outside the original function's domain.

Vertical asymptote of a logarithmic function

The boundary line occurring where the argument of the logarithmic function equals zero, represented by the line $x = -c$ for transformed functions of the form $y = a\log_b(x + c) + d$.