SOLVING LOGARITHMIC EQUATIONS

Note:

If you would like an in-depth review of logarithms, the rules of logarithms, logarithmic functions and logarithmic equations, click on logarithmic functions.

Solve for x in the following equation.

Problem 8.4d:

$\log _{.982}\left( x^{2}-6x+5\right) ^{2}=8$

Answer:

$x=3\pm \sqrt{4+\left( 0.982\right) ^{4}}$ and $x=3\pm \sqrt{4-\left( 0.982\right) ^{4}}.$ The approximate answers are $x\approx 0.779657509082,$ 1.2478358458, 4.75216415412, and $5.22034249092.\bigskip\bigskip$

Solution:

The above equation is valid only if the expression $\quad \left( x^{2}-6x+5\right) ^{2}>0.$ The expression $\left( x^{2}-6x+5\right) ^{2}>0$is valid when $x\neq 1$ and $x\neq 5.$Another way of saying this is that the domain is the set of real numbers where $x\neq 1\ or\ 5.$

Note: If, when solving the above problem, you simplify the logarithmic equation $\log _{.982}\left( x^{2}-6x+5\right) ^{2}=8$ to $\log _{.982}\left( x^{2}-6x+5\right) =4,$ you will lose half of your solutions. Why?

Recall that $\log _{.982}\left( x^{2}-6x+5\right) ^{2}$ is valid for all real numbers that are not equal to 1 or 5. However, $2\log _{.982}\left( x^{2}-6x+5\right)$ is valid for the set of real numbers $\left( -\infty ,1\right) \cup \left( 5,\infty \right) .$ This means that $\log _{.982}\left( x^{2}-6x+5\right) ^{2}=2\log _{.982}\left( x^{2}-6x+5\right)$only over the set of real numbers $\left( -\infty ,1\right) \cup \left( 5,\infty \right) .\bigskip$

So be careful when you simplify a logarithmic equation before solving it.

Convert the logarithmic equation $\log _{.982}\left( x^{2}-6x+5\right) ^{2}=8$ to an equivalent exponential equation with base 0.982.

$$\begin{eqnarray*}&& \\ \left( 0.982\right) ^{8} &=&\left( x^{2}-6x+5\right) ^{2... ...ft( 0.982\right) ^{4} &=&\pm \left( x^{2}-6x+5\right) \\ && \\ \end{eqnarray*}$$
$$\begin{eqnarray*}&& \\ x^{2}-6x+5 &=&\pm \left( 0.982\right) ^{4} \\ && \\ && \\ x^{2}-6x+5+4 &=&\pm \left( 0.982\right) ^{4}+4 \\ && \\ \end{eqnarray*}$$
$$\begin{eqnarray*}&& \\ \left( x-3\right) ^{2} &=&\pm \left( 0.982\right) ^{4}+4... ...& \\ x-3 &=&\pm \sqrt{4\pm \left( 0.982\right) ^{4}} \\ && \\ \end{eqnarray*}$$
$$\begin{eqnarray*}&& \\ x &=&3\pm \sqrt{4\pm \left( 0.982\right) ^{4}} \\ && \end{eqnarray*}$$

$$\begin{eqnarray*}x &=&3+\sqrt{4+\left( 0.982\right) ^{4}}\approx 5.22034249092 \... ...sqrt{4-\left( 0.982\right) ^{4}}\approx 4.75216415413 \\ && \\ \end{eqnarray*}$$
$$\begin{eqnarray*}&& \\ x &=&3-\sqrt{4+\left( 0.982\right) ^{4}}\approx 0.779657... ...3-\sqrt{4-\left( 0.982\right) ^{4}}\approx 1.24783584587 \\ && \end{eqnarray*}$$

The exact answers are $x=3+\sqrt{4+\left( 0.982\right) ^{4}},\ x=3+\sqrt{ 4-\left( 0.982\right) ^{4}},... ...-\sqrt{4+\left( 0.982\right) ^{4}},\ and\ x=3-\sqrt{4-\left( 0.982\right) ^{4}}$. The approximate answers are $x\approx 5.22034249092,\ x\approx 4.75216415413,\ x\approx 0.779657509082,\ and\ x\approx 1.24783584587.$

These answers may or may not be the solutions. You must check them numerically or graphically.

Numerical Check:

• Check the answer $x=3+\sqrt{4+\left( 0.982\right) ^{4}}\approx 5.22034249092$ by substituting 5.22034249092 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

  
  Left Side: $\qquad \log _{.982}\left( x^{2}-6x+5\right) ^{2}$
  

  $$\begin{eqnarray*}&& \\ &=&\displaystyle \frac{\ln \left( x^{2}-6x+5\right) ^{2}... ....982\right) } \\ && \\ &\approx &7.99999999896\approx 8 \\ && \end{eqnarray*}$$

  
  Right Side:$\qquad 8$

  
  Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 5.22034249092 for x, then $x\approx 5.22034249092$ is a solution.

  
  

• Check the answer $x=3+\sqrt{4-\left( 0.982\right) ^{4}}\approx 4.75216415413$ by substituting 4.75216415413 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

  
  Left Side: $\qquad \log _{.982}\left( x^{2}-6x+5\right) ^{2}$
  

  $$\begin{eqnarray*}&& \\ &=&\displaystyle \frac{\ln \left( x^{2}-6x+5\right) ^{2}... ....982\right) } \\ && \\ &\approx &7.99999999896\approx 8 \\ && \end{eqnarray*}$$

  
  Right Side:$\qquad 8$

  
  Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 4.75216415413 for x, then $x\approx 4.75216415413$ is a solution.

  
  

  
  

  
  

• Check the answer $x=3-\sqrt{4+\left( 0.982\right) ^{4}}\approx 0.779657509082$ by substituting 0.779657509082 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

  
  Left Side: $\qquad \log _{.982}\left( x^{2}-6x+5\right) ^{2}$
  

  $$\begin{eqnarray*}&& \\ &=&\displaystyle \frac{\ln \left( x^{2}-6x+5\right) ^{2}... ...82\right) } \\ && \\ &\approx &8.0000000000004\approx 8 \\ && \end{eqnarray*}$$
  
  

  
  Right Side:$\qquad 8$

  
  Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 0.779657509082 for x, then $x\approx 0.779657509082$ is a solution.

  
  

  
  

• Check the answer $x=3-\sqrt{4-\left( 0.982\right) ^{4}}\approx 1.24783584587$ by substituting 1.24783584587 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

  
  Left Side: $\qquad \log _{.982}\left( x^{2}-6x+5\right) ^{2}$
  

  $$\begin{eqnarray*}&& \\ &=&\displaystyle \frac{\ln \left( x^{2}-6x+5\right) ^{2}... ...82\right) } \\ && \\ &\approx &8.0000000000004\approx 8 \\ && \end{eqnarray*}$$

  
  Right Side:$\qquad 8$

  
  Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 1.24783584587 for x, then $x\approx 1.24783584587$ is a solution.

  
  

All four of the answers are the solutions to the original equation.

Graphical Check:

You can also check your answer by graphing $\quad f(x)=\log _{.982}\left( x^{2}-6x+5\right) ^{2}-8\quad$(formed by subtracting the right side of the original equation from the left side). Look to see where the graph crosses the x-axis; that will be the real solution. Note that the graph crosses the x-axis at four places: $0.779657509082,\ 1.24783584587,\ 4.75216415413$, and 5.22034249092. This means that $0.779657509082,\ 1.24783584587,\ 4.75216415413$, and 5.22034249092 are the real solutions.

If you have trouble graphing the function $f(x)=\log _{.982}\left( x^{2}-6x+5\right) ^{2}-8\quad$, graph the equivalent function $f(x)=\displaystyle \frac{ \ln \left( x^{2}-6x+5\right) ^{2}}{\ln \left( 0.982\right) }-8\quad$.

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