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171S5.5p Solving Exponential and Logarithmic Equations November 20, 2012
Solving Exponential Equations
MAT 171 Precalculus Algebra
Dr. Claude Moore
Cape Fear Community College
CHAPTER 5: Exponential and Logarithmic Functions
5.1 Inverse Functions
5.2 Exponential Functions and Graphs
5.3 Logarithmic Functions and Graphs
5.4 Properties of Logarithmic Functions
5.5 Solving Exponential and Logarithmic Equations 5.6 Applications and Models: Growth and Decay; and Compound Interest
5.5 Solving Exponential and
Logarithmic 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.
Base­Exponent Property
For any a > 0, a ≠ 1, ax = ay if and only if x = y.
Example
Solve
Solution:
Write each side as a power of the same number (base).
Since the bases are the same number, 2, we can use the base­exponent property and set the exponents equal:
Check x = 4:
• Solve exponential equations.
• Solve logarithmic equations.
TRUE
The solution is 4.
Nov 17­11:31 AM
Nov 17­11:31 AM
Example
Another Property
Solve: e0.08t = 2500.
Property of Logarithmic Equality
For any M > 0, N > 0, a > 0, and a ≠ 1,
loga M = loga N M = N.
Example
Solve: 3x = 20.
This is an exact answer. We cannot simplify further, but we can approximate using a calculator.
The solution is about 97.8.
Solving Logarithmic Equations
Equations containing variables in logarithmic expressions, such as
log2 x = 4 and log x + log (x + 3) = 1,
are called logarithmic equations.
We can check by finding 32.7268 ≈ 20.
To solve logarithmic equations algebraically, we first try to obtain a single logarithmic expression on one side and then write an equivalent exponential equation.
Nov 17­11:31 AM
Nov 17­11:31 AM
Example
Example
Solve: log3 x = −2.
Solve:
Check:
Solution:
TRUE
The solution is
Example
Only the value 2 checks, and it is the only solution.
Solve:
Solution:
Example ­ Using the Graphing Calculator
Solve: e0.5x – 7.3 = 2.08x + 6.2.
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.
Nov 17­11:31 AM
Solve:
Graph y1 = e0.5x – 7.3 and y2 = 2.08x + 6.2 and use the Intersect method.
The approximate solutions are –6.471 and 6.610.
Nov 17­11:31 AM
1
171S5.5p Solving Exponential and Logarithmic Equations Suggestions for solving exponential and logarithmic equations:
November 20, 2012
Solve the exponential equation algebraically. Then check using a graphing calculator.
452/2. 2x = 32
Exponential equation: Write so that bases are equal, set exponents equal, and solve, if possible. If not possible, write as logarithmic equation and solve.
Base­Exponent Property
For any a > 0, a ≠ 1, ax = ay if and only if x = y.
Logarithmic equation:
If bases of logs are equal, set quantities equal and solve. If not possible, write as exponential equation and solve.
Property of Logarithmic Equality: For any M > 0, N > 0, a > 0, and a ≠ 1,
loga M = loga N if and only if M = N.
Solve the exponential equation algebraically. Then check using a graphing calculator.
452/4. 37x = 27
loga x = y if and only if x = ay
Nov 18­6:48 AM
Solve the exponential equation algebraically. Then check using a graphing calculator.
452/6. 2x = 40
Solve the exponential equation algebraically. Then check using a graphing calculator.
452/10. Nov 17­12:40 PM
Nov 17­12:40 PM
Solve the exponential equation algebraically. Then check using a graphing calculator.
452/14. 15x = 30
Solve the exponential equation algebraically. Then check using a graphing calculator.
452/20. 1000e0.09t = 5000
Nov 20­9:07 AM
Solve the exponential equation algebraically. Then check using a graphing calculator.
452/24. 250 ­ (1.87) x = 0
Solve the logarithmic equation algebraically. Then check using a graphing calculator.
452/40. log5 (8 ­ 7x) = 3
Solve the exponential equation algebraically. Then check using a graphing calculator.
452/28. 2x+1 = 52x
2
171S5.5p Solving Exponential and Logarithmic Equations Solve the logarithmic equation algebraically. Then check using a graphing calculator.
444/48. log5 (x + 4) + log5 (x ­ 4) = 2
November 20, 2012
Use a graphing calculator to find the approximate solutions of the equation.
444/54. 0.082e0.05x = 0.034
Use a graphing calculator to find the approximate solutions of the equation.
444/62. log5 x + 7 = 4 ­ log5 x
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