Exponential and Logarithmic Equations

Objectives

This section covers methods for solving exponential and logarithmic equations, including expressing equations with like bases, using logarithms, applying the definition of a logarithm, and utilizing the one-to-one property. Applications such as compound interest and growth models are also discussed.

• Use like bases to solve exponential equations.

• Use logarithms to solve exponential equations.

• Apply the definition of a logarithm to solve logarithmic equations.

• Use the one-to-one property of logarithms to solve logarithmic equations.

• Solve applied problems involving exponential and logarithmic equations.

Solving Exponential Equations by Expressing Each Side as a Power of the Same Base

Exponential equations can often be solved by rewriting both sides with the same base, allowing the exponents to be set equal to each other.

• Key Point: If , then (for , ).

• Example: Solve by setting .

Using Logarithms to Solve Exponential Equations

When it is not possible to express both sides of an exponential equation with the same base, logarithms can be used to solve for the exponent.

• Key Point: Take the logarithm of both sides to isolate the exponent.

• Formula:

• Example: Solve by taking logarithms:

Using the Definition of a Logarithm to Solve Logarithmic Equations

The definition of a logarithm allows us to rewrite logarithmic equations in exponential form, which can then be solved algebraically.

• Key Point: If , then .

• Example: Solve by rewriting as .

Using the One-to-One Property of Logarithms to Solve Logarithmic Equations

Logarithmic equations with the same base can be solved by equating their arguments, using the one-to-one property.

• Key Point: If , then .

• Example: Solve by setting , so .

Applications of Exponential and Logarithmic Equations

Exponential and logarithmic equations are used in real-world applications such as compound interest and growth models.

• Compound Interest Formula:

• Example: How long will it take $1000 to grow to $3600 at 8% annual interest compounded quarterly?

• Solution: Set , , , , solve for : Divide both sides by 1000: Take logarithms: years

• Growth Model Example: At approximately age 14, a girl has attained 97% of her adult height. This can be modeled using exponential growth equations.

Summary Table: Methods for Solving Exponential and Logarithmic Equations