뒤로Exponential and Logarithmic Equations: Study Notes for College Algebra
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Exponential and Logarithmic Equations
Objectives
This section covers methods for solving exponential and logarithmic equations, including the use of like bases, logarithms, and properties of logarithmic functions. Applications to real-world problems 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
Method Overview
Exponential equations can often be solved by rewriting both sides of the equation with the same base. If , then .
Rewrite the equation so both sides have the same base.
Set the exponents equal to each other: .
Solve for the variable.
Example
Example: Solve .
Rewrite $8.
Set exponents equal: .
Solution set: {3}
Example: Solve .
Rewrite as .
Set exponents equal: .
Solution set: {-3}
Using Logarithms to Solve Exponential Equations
Method Overview
When the bases cannot be made the same, logarithms are used to solve exponential equations.
Isolate the exponential expression.
Take the logarithm of both sides (common logarithm for base 10, natural logarithm for other bases).
Simplify using properties:
Solve for the variable.
Example
Example: Solve .
Take logarithms:
Apply property:
Solve:
Solution set: {}
Using the Definition of a Logarithm to Solve Logarithmic Equations
Method Overview
Logarithmic equations can be solved by converting them to exponential form using the definition: means .
Express the equation in the form .
Rewrite in exponential form: .
Solve for the variable.
Check solutions in the original equation; only include values for which .
Example
Example: Solve .
Rewrite:
Solve:
Check: is true.
Solution set: {9}
Example: Solve .
Rewrite:
Solve:
Check: is true.
Solution set: {12}
Example: Solve .
Rewrite:
Solve:
Find solutions:
Check for positive values only.
Example: Solve .
Set arguments equal:
Solve:
Check: ,
Solution set: {1}
Example: Solve .
Set arguments equal:
Solve:
Check: ,
Solution set: {2.5}
Example: Solve .
Rewrite:
Solve:
Find solutions:
Check for positive values only.
Example: Solve .
Set arguments equal:
Solve:
Check: ,
Solution set: {2.5}
Using the One-to-One Property of Logarithms to Solve Logarithmic Equations
Method Overview
The one-to-one property states that if , then (provided and ).
Express the equation in the form .
Set .
Solve for the variable.
Check solutions in the original equation; only include values for which and .
Example
Example: Solve .
Set arguments equal:
Check:
Solution set: {5}
Example: Solve .
Set arguments equal:
Check:
Solution set: {4}
Example: Solve .
Set arguments equal:
Solve:
Check: ,
Solution set: {2.5}
Applications of Exponential and Logarithmic Equations
Compound Interest
Exponential equations are used to model compound interest. The formula for compound interest is:
A: Accumulated amount
P: Principal (initial amount)
r: Annual interest rate (decimal)
n: Number of compounding periods per year
t: Number of years
Example
Example: How long will it take $1000 to grow to $3600 at 8% annual interest compounded quarterly?
Set up:
Solve for using logarithms.
Solution: years
Growth Models
Logarithmic equations can model growth, such as the percentage of adult height attained by a child.
Example
Example: The percentage of adult height attained by a girl who is years old can be modeled by . At what age has a girl attained 97% of her adult height?
Set and solve for using logarithms.
Solution: years
