Algebra 2 Practice Test: Solving Logarithmic Equations
- Solving Logarithmic Equations
- 10 questions
- 15 minutes
These 10 questions practice solving logarithmic equations and exponential equations that need logarithms. A logarithm is defined only for positive inputs, so check every solution in the original equation.
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Question 1
Solve log5(x + 3) = 2.
Correct answer: B
Rewrite in exponential form: x + 3 = 52 = 25, so x = 22. Computing 5 · 2 = 10 instead of 52 gives x = 7.
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Question 2
Solve log4 x = 32.
Correct answer: A
x = 43/2 = (4)3 = 23 = 8. Multiplying 4 by 32 gives 6, which is not a power.
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Question 3
Solve log2(x + 6) − log2 x = 2.
Correct answer: D
Use the quotient rule: log2 x + 6x = 2, so x + 6x = 4. Then x + 6 = 4x and x = 2.
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Question 4
Solve log x + log(x − 3) = 1, where log means the common (base 10) logarithm.
Correct answer: C
The product rule gives x(x − 3) = 10, so x2 − 3x − 10 = 0 and x = 5 or x = −2. log(−2) is undefined, so only 5 is a solution.
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Question 5
Solve ln x = 3.
Correct answer: A
ln is the logarithm with base e, so ln x = 3 means x = e3 ≈ 20.09. 103 would be the answer for the common logarithm.
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Question 6
Solve e2x = 9.
Correct answer: B
Take ln of both sides: 2x = ln 9, so x = ln 92 = ln 91/2 = ln 3. Dividing 9 by 2 before taking the log gives ln 4.5.
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Question 7
What is the solution set of log3(x2) = 4?
Correct answer: C
x2 = 34 = 81, so x = 9 or x = −9. Both work, because (−9)2 = 81 is positive, so the logarithm is defined.
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Question 8
What is the solution set of 2 log7 x = log7 36?
Correct answer: B
The power rule gives log7 x2 = log7 36, so x2 = 36. In the original equation log7 x needs x > 0, so −6 is not allowed and only 6 remains.
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Question 9
Solve log6(x + 1) + log6(x − 4) = 1.
Correct answer: A
(x + 1)(x − 4) = 6, so x2 − 3x − 10 = 0 and x = 5 or x = −2. For x = −2 both logarithms have negative inputs, so only 5 works.
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Question 10
The pH of a solution is pH = −log[H+], where [H+] is the hydrogen ion concentration in moles per liter. What is [H+] for a solution with pH 4?
Correct answer: D
4 = −log[H+] means log[H+] = −4, so [H+] = 10−4 = 0.0001 mol/L. 0.0004 confuses 10−4 with 4 × 10−4.
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