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Wize High School Grade 12 Pre-Calculus Textbook > Logarithmic Functions

Solving Logarithmic Equations

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Solving Logarithmic Equations

A logarithmic equation is an equation containing one or more logarithms containing a variable in the argument of the logarithmic function.

It can be expressed as:
log⁡bx=k\boxed{{\log_{b}{x}=k}}
where k∈Rk\in\mathbb{R}.

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How to Solve Logarithmic Equations

  1. Combine like terms and express as a single logarithm.
  2. Exponentiate both sides of the equation.
  3. Solve for x.
  4. Determine if there are any extraneous solutions by verifying the answers.

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Example

log⁡2(x−1)=2\log_{2}(x-1)=2


Step 1.
Already completed.


Step 2/3.

log⁡2(x−1)=22log⁡2(x−1)=22x−1=4x=5\begin{array}{rcl} \log_{2}(x-1)&=&2\\\\ 2^{\log_{2}(x-1)}&=&2^2\\\\ x-1&=&4\\\\ x&=&5 \end{array}


Step 4.
Verify.

log⁡2(x−1)=2log⁡2(5−1)=2log⁡24=22=2✓\begin{array}{rcl} \log_{2}(x-1)&=&2\\\\ \log_{2}(5-1)&=&2\\\\ \log_{2}4&=&2\\\\ 2&=&2&&\color{red}\checkmark \end{array}


Therefore, x=5x=5 is the solution.
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Example: Solving Logarithmic Equations

Solve and check.

log⁡(x+2)+log⁡(x−1)=1\log(x+2)+\log(x-1)=1


log⁡(x+2)+log⁡(x−1)=1log⁡(x+2)(x−1)=1(x2+x−2)=101x2+x−2−10=0x2+x−12=0(x+4)(x−3)=0∴ x=−4,3\begin{array}{rcl} \log(x+2)+\log(x-1)&=&1\\\\ \log(x+2)(x-1)&=&1\\\\ (x^2+x-2)&=&10^1\\\\ x^2+x-2-10&=&0\\\\ x^2+x-12&=&0\\\\ (x+4)(x-3)&=&0\\\\ \therefore~x&=&-4, 3 \end{array}

Check.

log⁡(x+2)+log⁡(x−1)=1log⁡(3+2)+log⁡(3−1)=1log⁡(5)+log⁡(2)=1log⁡10=11=1✓\begin{array}{rcl} \log(x+2)+\log(x-1)&=&1\\\\ \log(3+2)+\log(3-1)&=&1\\\\ \log(5)+\log(2)&=&1\\\\ \log10&=&1\\\\ 1&=&1&&\color{red}\checkmark \end{array} log⁡(x+2)+log⁡(x−1)=1log⁡(−4+2)+log⁡(−4−1)=1log⁡(−2)+log⁡(−5)=1∴ x=−4  is an extraneous solution\begin{array}{rcl} \log(x+2)+\log(x-1)&=&1\\\\ \log(-4+2)+\log(-4-1)&=&1\\\\ \log(-2)+\log(-5)&=&1\\\\ \therefore~x&=&-4~~\text{is an extraneous solution} \end{array}


Therefore, x=3x=3 is the only solution.

Practice: Solving Logarithmic Equations

Solve and check.

log⁡5(x+1)+log⁡5(x−3)=1\log_{5}{(x+1)}+\log_{5}{(x-3)}=1

Practice: Solving Logarithmic Equations

Solve.

log⁡2x+log⁡4x=3\log_{2}{x}+\log_{4}{x}=3

Practice: Solving Logarithmic Equations

If log⁡41024=a+b\log_{4}{1024}=a+b and log⁡864=a−b\log_{8}{64}=a-b, what is aa and bb?
Extra Practice
Solving Logarithmic Equations
Solve log⁡2(2x+1)−log⁡2(x−1)=3\log_2{(2x+1)}-\log_2{(x-1)}=3.
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
If log⁡7343=b−a\log_{7}{343}=b-a and log⁡636=2a+2b\log_{6}{36}=2a+2b, what is aa and bb?
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
If log⁡5625=a+b\log_{5}{625}=a+b and log⁡327=2a−b\log_{3}{27}=2a-b, what is aa and bb?
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
Solve.
log⁡56+log⁡52x2=log⁡548 \log_5{6} + \log_5{2x^2}=\log_5{48}
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
Solve.
log⁡(x−3)−log⁡(x−5)=log⁡5\log(x-3)-\log(x-5)=\log5
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
Solve and check.
log⁡82+log⁡84x2=1\log_82+\log_84x^2=1
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
Solve and check.
log⁡9(x+6)−log⁡9x=log⁡92\log_9(x+6)-\log_9x=\log_92