1. Solving Exponential and Logarithmic Equations
Section 5.5
Solving Exponential and Logarithmic Equations

2. Objectives
Solve exponential equations.
Solve logarithmic equations.

3. Solving Exponential Equations
Equations with variables in the exponents, such as 3x = 20 and 25x = 64, are called exponential equations. Use the following property to solve exponential equations. Base-Exponent Property For any a > 0, a  1, ax = ay  x = y.

4. Example Solve: Write each side as a power of the same number (base).
Since the bases are the same number, 2, we can use the base-exponent property and set the exponents equal:
Check x = 4:
The solution is 4.
TRUE

5. Another Property
Property of Logarithmic Equality For any M > 0, N > 0, a > 0, and a  1, loga M = loga N  M = N.

6. Example
Solve: 3x = 20.
This is an exact answer. We cannot simplify further, but we can approximate using a calculator.
We can check by finding  20.

7. Example
Solve: 100e0.08t = 2500.
The solution is about 40.2.

8. Example
Solve: 4x+3 = 3-x.

9. Solving Logarithmic Equations
Equations containing variables in logarithmic expressions, such as
log2 x = and log x + log (x + 3) = 1,
are called logarithmic equations.
To solve logarithmic equations algebraically, we first try to obtain a single logarithmic expression on one side and then write an equivalent exponential equation.

10. Example
Solve: log3 x = 2.
Check:
TRUE
The solution is

11. Example
Solve:

12. Example (continued) Check x = 2: Check x = –5: FALSE TRUE
The number –5 is not a solution because negative numbers do not have real number logarithms. The solution is 2.

13. Example
Solve:
Only the value 2 checks and it is the only solution.

14. Example - Using the Graphing Calculator
Solve: e0.5x – 7.3 = 2.08x
Graph y1 = e0.5x – 7.3 and y2 = 2.08x and use the Intersect method.
The approximate solutions are –6.471 and