Logarithms & Exponential Equations What is a Log?

A logarithmic (log) function is the inverse of an exponential function. Comparison: We “divide” both sides to solve for in the equation , we “square root” both sides to solve for in the equation , and

we “log” to solve for in the exponential equation .

Exponential Function: Log Function:

Special Logarithms

Common Logarithm (log of base 10) Natural Logarithm (log of base e)

Properties of Logarithms

1. Examples: (a) , (b)

2.

Examples: (a) , (b)

3. Product Property: Examples: (a) , (b)

4. Quotient Property:

Examples: (a) , (b)

5. Power Property:

Examples: (a) , (b)

6. Inverse Properties: and

Examples: (a) , (b)

7. Change-of-Base Formula:

Examples: (a) , (b)

LSC-Montgomery Learning Center: Logarithms and Exponential Equations Page 2 Last Updated April 13, 2011

Solving Equations by Comparing Both Sides:

1. If , then .

2. If , then Solving Exponential Equations by Converting Both Sides to the Same Base:

Example 1: Solve for in the equation . Solution: Since , both sides of the equation can be converted to base 2.

(Bases equal, exponents then equal)

.

Example 2: Solve for in the equation .

Solution: Since and , both sides of the equation can be converted to base 3.

(Bases equal, exponents then equal)

Solving Exponential Equations by Taking Log on Both Sides: Example: Solve for in the equation . Solution: Since 2 and 3 cannot be converted to the same base, we must take log on both sides.

(Take natural log on both sides. Taking common log works as well) (Power Property) (Distribute )

(Collect like terms) (Factor )

.

Common Mistakes: Wrong: . Right: . Wrong: . Right: . Wrong: . Right: . Wrong: . Right: . Product Property.

Wrong: . Right: .

Wrong: . Right: . Cannot be simplified.