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Exponential and logarithmic equations practice problems

Solve exponential and logarithmic equations exactly, and reject the extraneous roots. 60 problems with worked solutions; in 60 of them every equation is proved by a computer algebra system.

13 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 13.112

medium
Solve \( \displaystyle 5^{2 x} - 4 \cdot 5^{x} + 3 = 0 \) exactly.
  1. Let u = 5ˣ: u² − 4u + 3 = 0 factors as (u − 3)(u − 1) = 0.
    Reviewed
  2. \[ \left. 5^{2 x} - 4 \cdot 5^{x} + 3 \right|_{\substack{ x=0 }} = 0 \]
    x = 0 satisfies the equation.✓ Proved
  3. \[ \left. 5^{2 x} - 4 \cdot 5^{x} + 3 \right|_{\substack{ x=\frac{\ln{\left(3 \right)}}{\ln{\left(5 \right)}} }} = 0 \]
    x = log(3)/log(5) satisfies the equation.✓ Proved
Answer \( x = 0,\ \frac{\ln{\left(3 \right)}}{\ln{\left(5 \right)}} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.110

easy
Solve \( \displaystyle 3^{3 x - 1} = 3 \) exactly.
  1. Write 3 as 3^1 and set the exponents equal.
    Reviewed
  2. \[ \left. 3^{3 x - 1} - 3 \right|_{\substack{ x=\frac{2}{3} }} = 0 \]
    x = 2/3 satisfies the equation.✓ Proved
Answer \( x = \frac{2}{3} \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.111

easy
Solve \( \displaystyle \ln_{3}\left(4 x + 2\right) = 1 \) exactly.
  1. Rewrite in exponential form: 4*x + 2 = 3^1.
    Reviewed
  2. \[ \left. \frac{\ln{\left(4 x + 2 \right)}}{\ln{\left(3 \right)}} - 1 \right|_{\substack{ x=1 \cdot \frac{1}{4} }} = 0 \]
    x = 1/4 satisfies the equation.✓ Proved
Answer \( x = \frac{1}{4} \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.113

easy
Solve \( \displaystyle 10^{2 x + 3} = 10 \) exactly.
  1. Write 10 as 10^1 and set the exponents equal.
    Reviewed
  2. \[ \left. 10^{2 x + 3} - 10 \right|_{\substack{ x=-1 }} = 0 \]
    x = -1 satisfies the equation.✓ Proved
Answer \( x = -1 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.118

easy
Solve \( \displaystyle 3^{x + 2} = 3 \) exactly.
  1. Write 3 as 3^1 and set the exponents equal.
    Reviewed
  2. \[ \left. 3^{x + 2} - 3 \right|_{\substack{ x=-1 }} = 0 \]
    x = -1 satisfies the equation.✓ Proved
Answer \( x = -1 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.181

easy
Solve \( \displaystyle 3^{x - 2} = 81 \) exactly.
  1. Write 81 as 3^4 and set the exponents equal.
    Reviewed
  2. \[ \left. 3^{x - 2} - 81 \right|_{\substack{ x=6 }} = 0 \]
    x = 6 satisfies the equation.✓ Proved
Answer \( x = 6 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.187

easy
Solve \( \displaystyle \ln_{3}\left(2 x - 5\right) = 3 \) exactly.
  1. Rewrite in exponential form: 2*x - 5 = 3^3.
    Reviewed
  2. \[ \left. \frac{\ln{\left(2 x - 5 \right)}}{\ln{\left(3 \right)}} - 3 \right|_{\substack{ x=16 }} = 0 \]
    x = 16 satisfies the equation.✓ Proved
Answer \( x = 16 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.188

easy
Solve \( \displaystyle 4 e^{x} = 8 \) exactly.
  1. Divide out the coefficient, then take ln of both sides.
    Reviewed
  2. \[ \left. 4 e^{x} - 8 \right|_{\substack{ x=\ln{\left(2 \right)} }} = 0 \]
    x = log(2) satisfies the equation.✓ Proved
Answer \( x = \ln{\left(2 \right)} \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.190

easy
Solve \( \displaystyle \ln_{10}\left(x + 5\right) = 1 \) exactly.
  1. Rewrite in exponential form: x + 5 = 10^1.
    Reviewed
  2. \[ \left. \frac{\ln{\left(x + 5 \right)}}{\ln{\left(10 \right)}} - 1 \right|_{\substack{ x=5 }} = 0 \]
    x = 5 satisfies the equation.✓ Proved
Answer \( x = 5 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.289

easy
Solve \( \displaystyle \ln_{2}\left(x - 4\right) = 2 \) exactly.
  1. Rewrite in exponential form: x - 4 = 2^2.
    Reviewed
  2. \[ \left. \frac{\ln{\left(x - 4 \right)}}{\ln{\left(2 \right)}} - 2 \right|_{\substack{ x=8 }} = 0 \]
    x = 8 satisfies the equation.✓ Proved
Answer \( x = 8 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.291

easy
Solve \( \displaystyle 10^{2 x} = 1000 \) exactly.
  1. Write 1000 as 10^3 and set the exponents equal.
    Reviewed
  2. \[ \left. 10^{2 x} - 1000 \right|_{\substack{ x=\frac{3}{2} }} = 0 \]
    x = 3/2 satisfies the equation.✓ Proved
Answer \( x = \frac{3}{2} \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.294

easy
Solve \( \displaystyle 2 e^{3 x} = 4 \) exactly.
  1. Divide out the coefficient, then take ln of both sides.
    Reviewed
  2. \[ \left. 2 e^{3 x} - 4 \right|_{\substack{ x=\frac{\ln{\left(2 \right)}}{3} }} = 0 \]
    x = log(2)/3 satisfies the equation.✓ Proved
Answer \( x = \frac{\ln{\left(2 \right)}}{3} \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.462

easy
Solve \( \displaystyle \ln_{3}\left(3 x\right) = 1 \) exactly.
  1. Rewrite in exponential form: 3*x = 3^1.
    Reviewed
  2. \[ \left. \frac{\ln{\left(3 x \right)}}{\ln{\left(3 \right)}} - 1 \right|_{\substack{ x=1 }} = 0 \]
    x = 1 satisfies the equation.✓ Proved
Answer \( x = 1 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More exponential and logarithmic equations practice

Solve \( \displaystyle \ln_{2}\left(4 x + 5\right) = 0 \) exactly.
Problem 13.106easy✓ Every equation proved
Solve \( \displaystyle 3 e^{3 x} = 9 \) exactly.
Problem 13.107easy✓ Every equation proved
Solve \( \displaystyle e^{2 x} = 5 \) exactly.
Problem 13.109easy✓ Every equation proved
Solve \( \displaystyle \ln_{10}\left(4 x + 2\right) = 1 \) exactly.
Problem 13.114easy✓ Nihil obstat
Solve \( \displaystyle \ln_{10}\left(3 x - 4\right) = 0 \) exactly.
Problem 13.115easy✓ Every equation proved
Solve \( \displaystyle \ln_{5}\left(2 x - 3\right) = 0 \) exactly.
Problem 13.119easy✓ Every equation proved
Solve \( \displaystyle e^{x} = 3 \) exactly.
Problem 13.120easy✓ Every equation proved
Solve \( \displaystyle 2^{3 x + 3} = 4 \) exactly.
Problem 13.182easy✓ Every equation proved
Solve \( \displaystyle 3 e^{3 x} = 21 \) exactly.
Problem 13.183easy✓ Every equation proved
Solve \( \displaystyle e^{2 x} = 3 \) exactly.
Problem 13.184easy✓ Every equation proved
Solve \( \displaystyle 2 e^{2 x} = 10 \) exactly.
Problem 13.189easy✓ Every equation proved
Solve \( \displaystyle 3 e^{x} = 9 \) exactly.
Problem 13.281easy✓ Nihil obstat
Solve \( \displaystyle 5^{3 x - 2} = 125 \) exactly.
Problem 13.282easy✓ Nihil obstat
Solve \( \displaystyle 3 e^{x} = 15 \) exactly.
Problem 13.284easy✓ Nihil obstat
Solve \( \displaystyle \ln_{5}\left(x - 1\right) = 1 \) exactly.
Problem 13.290easy✓ Nihil obstat
Solve \( \displaystyle 2^{x - 1} = 16 \) exactly.
Problem 13.292easy✓ Nihil obstat
Solve \( \displaystyle 2 e^{3 x} = 20 \) exactly.
Problem 13.381easy✓ Nihil obstat
Solve \( \displaystyle \ln_{5}\left(3 x + 5\right) = 2 \) exactly.
Problem 13.382easy✓ Nihil obstat
Solve \( \displaystyle 10^{x + 3} = 100 \) exactly.
Problem 13.383easy✓ Nihil obstat
Solve \( \displaystyle 2 e^{x} = 6 \) exactly.
Problem 13.385easy✓ Nihil obstat
Solve \( \displaystyle 5^{2 x + 1} = 25 \) exactly.
Problem 13.386easy✓ Every equation proved
Solve \( \displaystyle 2^{2 x + 3} = 4 \) exactly.
Problem 13.387easy✓ Every equation proved
Solve \( \displaystyle 5^{3 x - 1} = 5 \) exactly.
Problem 13.388easy✓ Nihil obstat
Solve \( \displaystyle 3^{3 x + 3} = 9 \) exactly.
Problem 13.389easy✓ Every equation proved
Solve \( \displaystyle e^{x} = 2 \) exactly.
Problem 13.460easy✓ Every equation proved
Solve \( \displaystyle 2 e^{4 x} = 6 \) exactly.
Problem 13.463easy✓ Nihil obstat
Solve \( \displaystyle 5^{x - 3} = 625 \) exactly.
Problem 13.464easy✓ Nihil obstat
Solve \( \displaystyle 2^{x - 3} = 2 \) exactly.
Problem 13.466easy✓ Nihil obstat
Solve \( \displaystyle 10^{3 x + 1} = 100 \) exactly.
Problem 13.467easy✓ Every equation proved
Solve \( \displaystyle \ln{\left(x \right)} + \ln{\left(x - 2 \right)} = \ln{\left(3 \right)} \) exactly.
Problem 13.108medium✓ Every equation proved
Solve \( \displaystyle \ln{\left(x \right)} + \ln{\left(x - 1 \right)} = \ln{\left(6 \right)} \) exactly.
Problem 13.116medium✓ Every equation proved
Solve \( \displaystyle 5^{2 x} - 13 \cdot 5^{x} + 36 = 0 \) exactly.
Problem 13.117medium✓ Every equation proved
Solve \( \displaystyle \ln{\left(x \right)} + \ln{\left(x - 2 \right)} = \ln{\left(15 \right)} \) exactly.
Problem 13.185medium✓ Every equation proved
Solve \( \displaystyle 3^{2 x} - 3 \cdot 3^{x} + 2 = 0 \) exactly.
Problem 13.186medium✓ Every equation proved
Solve \( \displaystyle 2^{2 x} - 13 \cdot 2^{x} + 36 = 0 \) exactly.
Problem 13.283medium✓ Every equation proved
Solve \( \displaystyle 5^{2 x} - 3 \cdot 5^{x} + 2 = 0 \) exactly.
Problem 13.285medium✓ Every equation proved
Solve \( \displaystyle 2^{2 x} - 10 \cdot 2^{x} + 9 = 0 \) exactly.
Problem 13.286medium✓ Every equation proved
Solve \( \displaystyle \ln{\left(x \right)} + \ln{\left(x - 1 \right)} = \ln{\left(2 \right)} \) exactly.
Problem 13.287medium✓ Every equation proved
Solve \( \displaystyle 3^{2 x} - 7 \cdot 3^{x} + 12 = 0 \) exactly.
Problem 13.288medium✓ Nihil obstat
Solve \( \displaystyle 10^{2 x} - 5 \cdot 10^{x} + 6 = 0 \) exactly.
Problem 13.293medium✓ Nihil obstat
Solve \( \displaystyle 10^{2 x} - 6 \cdot 10^{x} + 8 = 0 \) exactly.
Problem 13.295medium✓ Nihil obstat
Solve \( \displaystyle \ln{\left(x \right)} + \ln{\left(x - 4 \right)} = \ln{\left(32 \right)} \) exactly.
Problem 13.384medium✓ Every equation proved
Solve \( \displaystyle 5^{2 x} - 11 \cdot 5^{x} + 18 = 0 \) exactly.
Problem 13.390medium✓ Nihil obstat
Solve \( \displaystyle \ln{\left(x \right)} + \ln{\left(x - 4 \right)} = \ln{\left(21 \right)} \) exactly.
Problem 13.459medium✓ Every equation proved
Solve \( \displaystyle \ln{\left(x \right)} + \ln{\left(x - 3 \right)} = \ln{\left(28 \right)} \) exactly.
Problem 13.461medium✓ Every equation proved
Solve \( \displaystyle 3^{2 x} - 6 \cdot 3^{x} + 8 = 0 \) exactly.
Problem 13.465medium✓ Nihil obstat
Solve \( \displaystyle \ln{\left(x \right)} + \ln{\left(x - 2 \right)} = \ln{\left(24 \right)} \) exactly.
Problem 13.468medium✓ Every equation proved