Solving Equations Involving Logarithms
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When we have a variable in an exponent, we need to move it out of the exponent to determine it's value. To do that, we use the laws listed on the laws of logarithm page. Also, if the variable is inside a logarithm, we use exponentials to get it out of the logarithm. Here is how it works.
Next
Okay, after working these practice problems, you will be ready to tackle some application problems involving exponentials and logarithms, starting with exponential growth and decay.
Practice
Unless otherwise instructed, solve these problems using the exponential \(e\), giving your answers in exact terms.
Basic
\( \log_2(x+1) + \log_2(5x+1) = 6 \)
Problem Statement
Solve \( \log_2(x+1) + \log_2(5x+1) = 6 \)
Solution
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\( \ln x = 7 \)
Problem Statement
Solve \( \ln x = 7 \)
Solution
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\( \log_3(10x+1) - \log_3(x+1) = 2 \)
Problem Statement
Solve \( \log_3(10x+1) - \log_3(x+1) = 2 \)
Solution
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\( \log_2 16 = x \)
Problem Statement
Solve \( \log_2 16 = x \)
Solution
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\( \log_x 81 = 4 \)
Problem Statement
Solve \( \log_x 81 = 4 \)
Solution
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\( \log_{32} x = 4/5 \)
Problem Statement
Solve \( \log_{32} x = 4/5 \)
Solution
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\( \log_3(5x+1) = 4 \)
Problem Statement
Solve \( \log_3(5x+1) = 4 \)
Solution
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\( \log x = 24 \)
Problem Statement
Solve \( \log x = 24 \)
Solution
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\( \log_7 (x^2+3x+9) = 2 \)
Problem Statement
Solve \( \log_7 (x^2+3x+9) = 2 \)
Solution
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\( \ln(3x-2) = 5 \)
Problem Statement
Solve \( \ln(3x-2) = 5 \)
Solution
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\( 4\ln(2x-1) + 3 = 11 \)
Problem Statement
Solve \( 4\ln(2x-1) + 3 = 11 \)
Solution
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\( \log_3 (5x+2) = \log_3 (7x-8) \)
Problem Statement
Solve \( \log_3 (5x+2) = \log_3 (7x-8) \)
Solution
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\( \log_2 (x^2+4x) = \log_2 (5) \)
Problem Statement
Solve \( \log_2 (x^2+4x) = \log_2 (5) \)
Solution
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\( \log_2 x + \log_2 (x+4) = 5 \)
Problem Statement
Solve \( \log_2 x + \log_2 (x+4) = 5 \)
Solution
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\( \log_3 (x+1) = 3 - \log_3 (x=7) \)
Problem Statement
Solve \( \log_3 (x+1) = 3 - \log_3 (x=7) \)
Solution
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\( \log_4 (2x+6) - \log_4 (x-1) = 1 \)
Problem Statement
Solve \( \log_4 (2x+6) - \log_4 (x-1) = 1 \)
Solution
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\( \log_2 (x+3) = 4 + \log_2 (x-3) \)
Problem Statement
Solve \( \log_2 (x+3) = 4 + \log_2 (x-3) \)
Solution
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Advanced
\( \log x^{\log x} = 49 \)
Problem Statement
Solve \( \log x^{\log x} = 49 \)
Solution
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\( \log x^2 = (\log x)^2 \)
Problem Statement
Solve \( \log x^2 = (\log x)^2 \)
Solution
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\( \log (\log x) = 4 \)
Problem Statement
Solve \( \log (\log x) = 4 \)
Solution
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