뒤로Solving Exponential and Logarithmic Equations (Precalculus Study Guide)
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Section 3.4: Solving Exponential and Logarithmic Equations
Exponential Equations
Exponential equations are equations in which the variable appears in the exponent. Solving these equations often involves expressing both sides with the same base or using logarithms to isolate the variable.
Definition: An exponential equation is an equation of the form or where the variable is in the exponent.
Key Methods:
Expressing Both Sides as Powers of the Same Base: If , then .
Using Logarithms: If the bases cannot be made the same, take the logarithm (common or natural) of both sides after isolating the exponential expression.
Solving by Expressing Both Sides as Powers of the Same Base
Example 1: Solve
Rewrite $125.
Now so .
Solution set:
Example 2: Solve
Rewrite and .
Solution set:
Solving by Taking Logarithms
Example 3: Solve
Take logarithm of both sides:
Alternatively, use natural log:
Solution set: or
Example 4: Solve
Isolate exponential:
Take natural log:
Example 5: Solve
Take natural log:
Expand:
Rearrange:
Solution set:
Example 6: Solve
Let , so
Factor: or
;
Solution set:
Logarithmic Equations
Logarithmic equations contain variables inside logarithmic expressions. These equations are solved by rewriting them in exponential form or using logarithmic properties.
Definition: A logarithmic equation is an equation involving logarithms, such as or .
Key Methods:
Rewriting in Exponential Form: If , then .
Using Logarithmic Properties: Combine logs using product, quotient, or power rules before solving.
One-to-One Property: If , then .
Domain Considerations: Only include solutions where the argument of every logarithm is positive.
Solving by Rewriting in Exponential Form
Example 1: Solve
Rewrite:
Check: ; valid.
Solution set:
Example 2: Solve
Check: ; valid.
Solution set:
Example 3: Solve
Product rule:
Rewrite:
Factor: or
Check: is valid (, ); is not valid (logarithm of negative).
Solution set:
Solving by One-to-One Property
Property: If , then (provided and ).
Example: Solve
Quotient rule:
Set arguments equal:
Multiply both sides by :
Expand:
Rearrange:
Factor: or
Check: Both and make all arguments positive.
Solution set:
Application: Population Models
Exponential equations are commonly used to model population growth. The general form is , where is the population at time , is the initial population, and is the growth rate.
Example: The formula models the population of Florida, (in millions), years after 2000.
(a) Population in 2000: ; million.
(b) When will the population reach 19.2 million?
Set :
Divide:
Take natural log:
So, about 8 years after 2000, i.e., in 2008.
Summary Table: Methods for Solving Exponential and Logarithmic Equations
Equation Type | Method | Key Steps | Example |
|---|---|---|---|
Exponential (same base) | Express both sides with same base | Set exponents equal | |
Exponential (different base) | Take logarithm of both sides | Isolate exponent, apply log, solve for variable | |
Logarithmic (single log) | Rewrite in exponential form | Set argument equal to base raised to exponent | |
Logarithmic (multiple logs) | Combine logs, rewrite in exponential form | Use log properties, solve resulting equation | |
Logarithmic (one-to-one) | Set arguments equal | Check domain, solve for variable |
Additional info: All solutions to logarithmic equations must be checked to ensure the arguments of all logarithms are positive, as logarithms of zero or negative numbers are undefined.
