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Solving logarithmic equations worksheets Grade 11

Combine the logs, rewrite without logs, solve, and check each answer.

The front of the free solving logarithmic equations worksheet
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Free · Worked example · 8 problems · Answer key · US Letter

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How to do it

Combine the logs into one, rewrite without logs, and solve. Check each answer: you cannot take the log of 0 or a negative number.

log⁡3x+log⁡3(x+26)=3x=1\log_{3} x + \log_{3} (x + 26) = 3\quad x = 1
x(x+26)=27x(x + 26) = 27adding logs: multiply inside; then 3³ = 27
x2+26x−27=0x^{2} + 26x - 27 = 0move every term to one side
(x−1)(x+27)=0(x - 1)(x + 27) = 0factor
x=1 or x=−27x = 1\ \text{or }x = -27set each factor equal to 0
x=−27 would take the log⁡ of a negative number, so it is not a solution.x = -27\ \text{would take the }\log \ \text{of a negative number, so it is not a solution.}check each answer
x=1x = 1

Example problems

  1. log⁡2x+log⁡2(x+30)=6\log_{2} x + \log_{2} (x + 30) = 6
  2. log⁡3x+log⁡3(x+8)=2\log_{3} x + \log_{3} (x + 8) = 2
  3. log⁡2(x+4)−log⁡2x=log⁡25\log_{2} (x + 4) - \log_{2} x = \log_{2} 5
  4. log⁡2(x+16)−log⁡2x=log⁡23\log_{2} (x + 16) - \log_{2} x = \log_{2} 3

Exponential and logarithmic functions

  1. Solving exponential equations with a common base
  2. Evaluating logarithms
  3. Solving logarithmic equations with the definition
  4. Logarithm rules with numbers
  5. Expanding and condensing logarithms
  6. Change of base formula
  7. Solving exponential equations with logarithms
  8. Solving logarithmic equations
  9. Exponential growth and decay