Exponential and Logarithmic Equations: Definitions, Properties, and Solution Methods

Section 5.4 Exponential and Logarithmic Equations

Definition of the Logarithmic Function

The logarithmic function is a fundamental concept in precalculus, providing a way to solve equations involving exponents by rewriting them in logarithmic form.

• Definition: For $x > 0$, $a > 0$, and $a \neq 1$, $y = \log_a x$ if and only if $x = a^y$.

• Application: Logarithms are used to solve for exponents in equations where the variable is in the exponent.

Logarithm Property of Equality

This property allows us to equate the arguments of two logarithms with the same base.

• If $\log_a x = \log_a y$, then $x = y$.

Properties of Logarithms

Logarithms have several key properties that simplify expressions and solve equations.

• Let $a > 0$, $a \neq 1$, and $p, q, r$ be any real numbers.

• Product Rule: $\log_a(pq) = \log_a p + \log_a q$

• Quotient Rule: $\log_a\left(\frac{p}{q}\right) = \log_a p - \log_a q$

• Power Rule: $\log_a(p^r) = r \log_a p$

Change of Base Formula

The change of base formula allows us to rewrite logarithms in terms of logarithms with a different base, often base 10 or base $e$ for calculator use.

• For any positive base $a \neq 1$ and any positive real number $x$, $\log_a x = \frac{\log_b x}{\log_b a}$, where $b$ is any positive base not equal to 1.

Review of Solving Exponential Equations with Negative and Rational Exponents

• Exponential equations may involve negative or rational exponents, which can be solved using logarithms and properties of exponents.

Review of Solving Quadratic Equations by Factoring

• Some exponential equations can be transformed into quadratic form and solved by factoring.

Review of Solving Rational Equations

• Rational equations may arise in the context of logarithmic and exponential equations and can be solved using algebraic techniques.

Review of Solving Exponential Equations by Relating the Bases

When both sides of an exponential equation can be written with the same base, the exponents can be set equal to each other.

• If $a^x = a^p$, then $x = p$.

Objective 1: Solving Exponential Equations

Exponential equations are equations in which the variable appears in the exponent. There are systematic methods for solving these equations.

• If the equation can be written in the form $A^x = a$, then solve $x = \log_A a$.

• If the equation can be written as $A^{f(x)} = c$, where $c$ is a constant not equal to any power of $A$:

  • Rewrite the equation in logarithmic form using the definition of a logarithmic function.

  • Solve for the given variable and use the change of base formula if necessary.

• If the equation cannot be written in the form $A^x = a$:

  • Use the logarithm property of equality to "take the log of both sides" (often $\log$ or $\ln$).

  • Use the power rule of logarithms to "bring down" any exponents.

  • Solve for the given variable.

Example: Solve $2^x = 7$.

• Take logarithms of both sides: $\log(2^x) = \log(7)$

• Apply the power rule: $x \log(2) = \log(7)$

• Solve for $x$: $x = \frac{\log(7)}{\log(2)}$

Objective 2: Solving Logarithmic Equations

Logarithmic equations are equations in which the variable appears inside a logarithm. These can often be solved by rewriting the equation in exponential form or by combining logarithms.

• If the equation can be written in the form $\log_a x = r$, then solve $x = a^r$.

• If the equation cannot be written in this form:

  • Use properties of logarithms to combine all logarithms and write as a single logarithm if possible.

  • Use the definition of a logarithmic function to rewrite the equation in exponential form.

  • Solve for the given variable.

  • Check for any extraneous solutions. Verify that each solution results in the arguments of all logarithms in the original equation being greater than zero.

Example: Solve $\log_3(x-1) + \log_3(x+1) = 2$.

• Combine using the product rule: $\log_3[(x-1)(x+1)] = 2$

• Rewrite in exponential form: $(x-1)(x+1) = 3^2 = 9$

• Solve: $x^2 - 1 = 9 \implies x^2 = 10 \implies x = \pm \sqrt{10}$

• Check for extraneous solutions: Both $x = \sqrt{10}$ and $x = -\sqrt{10}$ must be checked to ensure the arguments of the logarithms are positive.

Important: Always verify solutions to logarithmic equations, as the process may produce extraneous solutions.

Summary Table: Properties of Logarithms

Property	Formula	Description
Product Rule	$\log_a(pq) = \log_a p + \log_a q$	Logarithm of a product equals the sum of the logarithms.
Quotient Rule	$\log_a\left(\frac{p}{q}\right) = \log_a p - \log_a q$	Logarithm of a quotient equals the difference of the logarithms.
Power Rule	$\log_a(p^r) = r \log_a p$	Logarithm of a power equals the exponent times the logarithm.
Change of Base	$\log_a x = \frac{\log_b x}{\log_b a}$	Allows conversion between logarithms of different bases.

Additional info: These notes cover the essential methods for solving exponential and logarithmic equations, including the use of logarithmic properties, change of base, and verification of solutions. Mastery of these techniques is crucial for success in precalculus and further studies in mathematics.