Exponential and Logarithmic Functions - Tahsili Review

Vocabulary practice flashcards covering fundamental definitions, formulas, and properties of exponential and logarithmic functions from the 3rd Secondary Tahsili review booklet.

Exponential Decay Function ($f(x) = b^x$)

An exponential function where the base $b$ satisfies the condition $0 < b < 1$.

Logarithmic to Exponential Form Equivalence

The relationship stating that a logarithmic equation $\log_a(b) = c$ is mathematically equivalent to the exponential equation $a^c = b$.

Product Property of Logarithms

A property stating that the sum of two logarithms with identical bases simplifies to the logarithm of their product: $\log_b(x) + \log_b(y) = \log_b(xy)$.

Quotient Property of Logarithms

A property stating that the difference between two logarithms with identical bases simplifies to the logarithm of the quotient: $\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)$.

Power Property of Logarithms

A property stating that a multiplier in front of a logarithm can be rewritten as an exponent of the argument: $c\log_b(x) = \log_b(x^c)$.

Vertical Asymptote of a Logarithmic Function

A vertical line of the form $x = h$ approached by the curve of $f(x) = \log_b(x - h)$, found by setting the argument of the logarithm equal to zero ($x - h = 0$).

Zero of a Function

The input value $x$ for which the function's output evaluates to zero ($f(x) = 0$).

$y$-intercept

The value of the dependent variable $y$ where the graph of a function intersects the $y$-axis, evaluated algebraically by setting $x = 0$.

Domain of a Basic Logarithmic Function ($f(x) = \log_b(x)$)

The set of all positive real numbers ($x > 0$), meaning non-positive numbers ($x \le 0$) cannot belong to the domain.

Inverse Function of a Logarithmic Function

An exponential function obtained by switching the variables and solving for $y$, mapping $f(x) = \log_b(x + c)$ to $f^{-1}(x) = b^x - c$.

Base Equality Property for Exponential Equations

The algebraic property stating that if $b^u = b^v$ for a positive real base $b \ne 1$, then the exponents must be equal: $u = v$.

Sound Level ($dB$) Formula

A logarithmic intensity formula expressed as $dB = 10\log\left(\frac{I}{I_0}\right)$, where $I$ is the sound intensity and $I_0$ is the threshold intensity of hearing.

Common Logarithm

A logarithm with base $10$, denoted without an explicit base subscript as $\log(x)$, where $\log(10) = 1$ and $\log(100) = 2$.

Logarithmic Inequality Domain Restriction

The condition requiring that any prospective solution to $\log_b(f(x)) < \log_b(g(x))$ must strictly ensure that both arguments remain positive: $f(x) > 0$ and $g(x) > 0$.