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Algebra and Precalculus A Practical Guide

Cameron Geisler

Section 13.7 Solving Logarithmic Equations

Subsection 13.7.1 Examples

Exercise Group 13.7.1. Basic Equations.

Exercise Group 13.7.2. Intermediate Equations.

Solve each equation.
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(a)
\(\log_{3}(9x) + \log_{3}x = 4\)
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Answer.
\(x = 3\) (reject \(x = -3\))
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(b)
\(\log_{3} x + \log_{3}(x-4) = \log_3 12\)
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Answer.
\(x = 6\) (reject \(x = -2\))
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(c)
\(\log_5{(x-1)}+\log_5{3}=\log_5{x}\)
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Answer.
\(x=\frac{3}{2}\)
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(d)
\(\log_{6}(x+3)+\log_{6}(x+4)=1\)
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Answer.
\(x=-1\) (reject \(x=-6\))
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(e)
\(2 \log_3{x}-\log_3(x+2) = 0\)
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Answer.
\(x=2\) (reject \(x=-1\))
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(f)
\(\log_2(x-3)+\log_2{x}=2\)
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Answer.
\(x=4\) (reject \(x=-1\))
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(h)
\(\log_3{(x+6)}-\log_3{(x+2)}=\log_3{x}\)
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Answer.
\(x = 2\) (reject \(x = -3\))
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(j)
\(\log(2x+4)-\log(x+2)=\log(x+1)\)
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Answer.
\(x=1\) (reject \(x=-2\))
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(k)
\(2\log_{2}(x-4)-\log_2{x}=1\)
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Answer.
\(x=8\) (reject \(x=2\))
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(l)
\(\log(6x)=\log(x+6)+\log(x-1)\)
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Answer.
\(x=3\) (reject \(x=-2\))
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(m)
\(\log_4(x+7)=2-\log_4(x+1)\)
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Answer.
\(x=1\) (reject \(x = -9\))
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(n)
\(\log_2{(2x+4)}-\log_2{(x-1)}=\log_3{27}\)
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Answer.
\(x=2\)
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(o)
\(\log{(x+2)}+\log{(x-1)}=1\)
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Answer.
\(x=3\) (reject \(x=-4\))
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(p)
\(\log_{15}{(x+2)}+\log_{15}{x}=1\)
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Answer.
\(x=3\) (reject \(x=-5\))
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(q)
\(\log_{3}{(x+3)}=1-\log_{3}{(x+5)}\)
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Answer.
\(x=-2\) (reject \(x=-6\))
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(r)
\(\log{(x-3)}+\log{(x+4)}-\log{x}=\log{5}\)
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Answer.
\(x=6\) (reject \(x=-2\))
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(t)
\(\log_3(x+6)-\log_3(x+2)=\log_3{x}\)
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Answer.
\(x=2\) (reject \(x=-3\))
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Exercise Group 13.7.3. Mixed Equations.

Solve each equation.
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(b)
\(\log_5(2x-1)+\log_5(x-2)=1\)
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Answer.
\(x=3\) (reject \(x=-\frac{1}{2}\))
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(d)
\(\log_5{(x-18)}-\log_5{x}=\log_5{7}\)
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Answer.
no solution (reject \(x=-3\))
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(e)
\(2\log_3{(3-x)} = \log_3{4} + \log_3{(6-x)}\)
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Answer.
\(x = -3\) (reject \(x = 5\))
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(f)
\(\log(2x-3)+\log(x-2)=\log(2x-1)\)
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Answer.
\(x=\frac{7}{2}\) (reject \(x=1\))
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(g)
\(\log(x-4)+\log(x+1)=\log(x-8)\)
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Answer.
no solution (reject \(x=2\))
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(h)
\(\log_2(2-2x)+\log_2(1-x)=5\)
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Answer.
\(x=-3\) (reject \(x=5\))
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(i)
\(\log_{5}(3x+1)+\log_{5}(x-3)=3\)
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Answer.
\(x=8\) (reject \(x=-\frac{16}{3}\))
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(j)
\(\log_2(3x+1)+\log_2(x-1)=\log_2(10x+14)\)
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Answer.
\(x=5\) (reject \(x=-1\))
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(k)
\(2\log(4-x)-\log{3}=\log(10-x)\)
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Answer.
\(x=-2\) (reject \(x=7\))
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(l)
\(\log_2{(x-3)}+\log_2{(x-2)}=2+\log_2{5}\)
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Answer.
\(x=7\) (reject \(x=-2\))
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(n)
\(\log_{2}{(x^2-12)}-\log_{2}{1}=\log_{2}{x}\)
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Answer.
\(x=4\) (reject \(x=-3\))
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(p)
\(2\log_4{x}+\log_4{(x-2)}-\log_4{(2x)}=1\)
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Answer.
\(x=4\) (reject \(x=-2\))
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(q)
\(2\log_3{x}+\log_3{(x-1)}=1+\log_3{(2x)}\)
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Answer.
\(x=3\) (reject \(x=-2\))
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Exercise Group 13.7.4. Advanced Examples.

Solve each equation.
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(a)
\(\log_2{(x+1)} + \log_2{x} = \log_2{5}\)
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Hint.
quadratic equation, solve using the quadratic formula
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Answer.
\(x = \frac{-1+\sqrt{21}}{2} \approx 1.79\) (reject \(x = \frac{-1-\sqrt{21}}{2} \approx -2.79\))
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(b)
\(\log_4(3x^2-5x-2)-\log_4(x-2)=1\)
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Answer.
no solution (reject \(x=1,2\))
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(c)
\(\log_{25}(x-1)+\log_{25}(x+3)=\log_7\sqrt{7}\)
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Answer.
\(x=2\) (reject \(x=-4\))
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(d)
\(\frac{1}{2}-\log_{16}(x-3)=\log_{16}x\)
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Answer.
\(x=4\) (reject \(x=-1\))
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(e)
\(\log_{36}(x-2)+\log_{36}(x+1)+\log_{36}(x-3)=\frac{1}{2}\)
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Hint.
Factor the common factor, and then use the quadratic formula
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Answer.
\(x=2+\sqrt{3}\) (reject \(x=0,2-\sqrt{3}\))
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(f)
\(\log_{1/4}(x^2+2)-\log_{1/4}(x^2-x)=-2\)
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Hint.
\(15x^2-16x-2=0\text{,}\) use the quadratic formula
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Answer.
\(x = \frac{8 \pm \sqrt{94}}{15} \approx 1.18, -0.11\)
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