Exponential and Logarithmic Equations: Mini-Quiz Review

Exponential Equations

Solving Exponential Equations with the Same Base

Exponential equations can often be solved by expressing both sides with the same base, then equating exponents.

• Key Point: If $a^x = a^y$, then $x = y$ (provided $a \neq 0, 1$).

• Example: Solve $2^{3x+2} = \left(\frac{1}{4}\right)^{x-2}$.

Rewrite $\frac{1}{4}$ as $2^{-2}$:

$2^{3x+2} = (2^{-2})^{x-2}$

$2^{3x+2} = 2^{-2(x-2)}$

Set exponents equal:

$3x + 2 = -2(x-2)$

$3x + 2 = -2x + 4$

$5x = 2$

$x = \frac{2}{5}$

Solving Exponential Equations with Different Bases

Sometimes, bases must be rewritten to a common base before equating exponents.

• Example: $3^{x-2} = 27^{x-5}$

Rewrite $27$as $3^3$:

$3^{x-2} = (3^3)^{x-5}$

$3^{x-2} = 3^{3x-15}$

Set exponents equal:

$x-2 = 3x-15$

$2x = 13$

$x = \frac{13}{2}$

Graphing Exponential Functions

Key Features of Exponential Graphs

• Key Point: The general form is $y = a \cdot b^{x} + k$.

• Key Features:

  • Key Point (KP): The y-intercept, found by setting $x=0$.

  • Asymptote: The horizontal line $y = k$ that the graph approaches but never crosses.

• Example: $y = 3 \cdot 2^{x-2} - 3$

  • KP: $(0, 3 \cdot 2^{-2} - 3) = (0, -2.25)$

  • Asymptote: $y = -3$

Logarithmic Equations

Solving Logarithmic Equations

Logarithmic equations can be solved by rewriting them in exponential form or by using properties of logarithms.

• Key Point: $\log_b(a) = c \iff b^c = a$

• Example: $\log_3 x = 4$

Rewrite in exponential form:

$3^4 = x$

$x = 81$

• Example: $\log_{10} x = -5$

$10^{-5} = x$

$x = 0.00001$

Solving Logarithmic Equations with Exponents

• Example: $\log_{3} x^3 = 27$

Rewrite $27$as $3^3$:

$\log_{3} x^3 = 3$

$x^3 = 3^3$

$x = 3$

Solving Equations with Rational Exponents

Solving for x with Rational Exponents

Equations involving rational exponents can be solved by raising both sides to the reciprocal power.

• Key Point: $x^{m/n} = a \implies x = a^{n/m}$

• Example: $x^{2/3} = 25$

Raise both sides to the $3/2$ power:

$(x^{2/3})^{3/2} = 25^{3/2}$

$x = \pm 125$

• Example: $(2x+4)^{3/2} = 64$

Raise both sides to the $2/3$ power:

$(2x+4)^{3/2 \cdot 2/3} = 64^{2/3}$

$2x+4 = 16$

$x = 6$

Summary Table: Key Properties of Exponential and Logarithmic Equations

Equation Type	Key Property	Solution Method
Exponential (same base)	$a^x = a^y \implies x = y$	Rewrite bases, set exponents equal
Logarithmic	$\log_b(a) = c \iff b^c = a$	Rewrite in exponential form
Rational Exponents	$x^{m/n} = a \implies x = a^{n/m}$	Raise both sides to reciprocal power

Additional info: The notes also include sketches of exponential graphs, highlighting key points and asymptotes, and emphasize the importance of rewriting expressions to common bases or forms for easier solving.