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Algebra
Logarithms
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Solve \(\displaystyle \log_{ 8 }(x + 12) + \log_{ 8 }(x + 516) = 4\)

Using the product rule for logarithms gives \(\displaystyle \log_{ 8 }(\left(x + 12\right) \left(x + 516\right)) \) and rewriting in exponential form gives \(\displaystyle \left(x + 12\right) \left(x + 516\right) = 4096\) expanding and setting the equation equal to zero gives \(\displaystyle x^{2} + 528 x + 2096 = 0\). Factoring gives \(\displaystyle \left(x + 4\right) \left(x + 524\right)=0\). This gives two possible solutions \(\displaystyle x=-524\) or \(\displaystyle x=-4\). \(\displaystyle x=-524\) is an extraneous solution. The only soution is \(\displaystyle x=-4\).

Download \(\LaTeX\) 

\begin{question}Solve $\log_{ 8 }(x + 12) + \log_{ 8 }(x + 516) = 4$
    \soln{10cm}{Using the product rule for logarithms gives $\log_{ 8 }(\left(x + 12\right) \left(x + 516\right)) $ and rewriting in exponential form gives $\left(x + 12\right) \left(x + 516\right) = 4096$ expanding and setting the equation equal to zero gives $x^{2} + 528 x + 2096 = 0$.  Factoring gives $\left(x + 4\right) \left(x + 524\right)=0$. This gives two possible solutions $x=-524$ or $x=-4$. $x=-524$ is an extraneous solution.  The only soution is $x=-4$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Solve  <img class="equation_image" title=" \displaystyle \log_{ 8 }(x + 12) + \log_{ 8 }(x + 516) = 4 " src="/equation_images/%20%5Cdisplaystyle%20%5Clog_%7B%208%20%7D%28x%20%2B%2012%29%20%2B%20%5Clog_%7B%208%20%7D%28x%20%2B%20516%29%20%3D%204%20" alt="LaTeX:  \displaystyle \log_{ 8 }(x + 12) + \log_{ 8 }(x + 516) = 4 " data-equation-content=" \displaystyle \log_{ 8 }(x + 12) + \log_{ 8 }(x + 516) = 4 " /> </p> </p>
HTML for Canvas
<p> <p>Using the product rule for logarithms gives  <img class="equation_image" title=" \displaystyle \log_{ 8 }(\left(x + 12\right) \left(x + 516\right))  " src="/equation_images/%20%5Cdisplaystyle%20%5Clog_%7B%208%20%7D%28%5Cleft%28x%20%2B%2012%5Cright%29%20%5Cleft%28x%20%2B%20516%5Cright%29%29%20%20" alt="LaTeX:  \displaystyle \log_{ 8 }(\left(x + 12\right) \left(x + 516\right))  " data-equation-content=" \displaystyle \log_{ 8 }(\left(x + 12\right) \left(x + 516\right))  " />  and rewriting in exponential form gives  <img class="equation_image" title=" \displaystyle \left(x + 12\right) \left(x + 516\right) = 4096 " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28x%20%2B%2012%5Cright%29%20%5Cleft%28x%20%2B%20516%5Cright%29%20%3D%204096%20" alt="LaTeX:  \displaystyle \left(x + 12\right) \left(x + 516\right) = 4096 " data-equation-content=" \displaystyle \left(x + 12\right) \left(x + 516\right) = 4096 " />  expanding and setting the equation equal to zero gives  <img class="equation_image" title=" \displaystyle x^{2} + 528 x + 2096 = 0 " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%20%2B%20528%20x%20%2B%202096%20%3D%200%20" alt="LaTeX:  \displaystyle x^{2} + 528 x + 2096 = 0 " data-equation-content=" \displaystyle x^{2} + 528 x + 2096 = 0 " /> .  Factoring gives  <img class="equation_image" title=" \displaystyle \left(x + 4\right) \left(x + 524\right)=0 " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28x%20%2B%204%5Cright%29%20%5Cleft%28x%20%2B%20524%5Cright%29%3D0%20" alt="LaTeX:  \displaystyle \left(x + 4\right) \left(x + 524\right)=0 " data-equation-content=" \displaystyle \left(x + 4\right) \left(x + 524\right)=0 " /> . This gives two possible solutions  <img class="equation_image" title=" \displaystyle x=-524 " src="/equation_images/%20%5Cdisplaystyle%20x%3D-524%20" alt="LaTeX:  \displaystyle x=-524 " data-equation-content=" \displaystyle x=-524 " />  or  <img class="equation_image" title=" \displaystyle x=-4 " src="/equation_images/%20%5Cdisplaystyle%20x%3D-4%20" alt="LaTeX:  \displaystyle x=-4 " data-equation-content=" \displaystyle x=-4 " /> .  <img class="equation_image" title=" \displaystyle x=-524 " src="/equation_images/%20%5Cdisplaystyle%20x%3D-524%20" alt="LaTeX:  \displaystyle x=-524 " data-equation-content=" \displaystyle x=-524 " />  is an extraneous solution.  The only soution is  <img class="equation_image" title=" \displaystyle x=-4 " src="/equation_images/%20%5Cdisplaystyle%20x%3D-4%20" alt="LaTeX:  \displaystyle x=-4 " data-equation-content=" \displaystyle x=-4 " /> . </p> </p>