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Algebra
Logarithms
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Solve \(\displaystyle \log_{ 4 }(x + 6) + \log_{ 4 }(x + 66) = 4\)

Using the product rule for logarithms gives \(\displaystyle \log_{ 4 }(\left(x + 6\right) \left(x + 66\right)) \) and rewriting in exponential form gives \(\displaystyle \left(x + 6\right) \left(x + 66\right) = 256\) expanding and setting the equation equal to zero gives \(\displaystyle x^{2} + 72 x + 140 = 0\). Factoring gives \(\displaystyle \left(x + 2\right) \left(x + 70\right)=0\). This gives two possible solutions \(\displaystyle x=-70\) or \(\displaystyle x=-2\). \(\displaystyle x=-70\) is an extraneous solution. The only soution is \(\displaystyle x=-2\).

Download \(\LaTeX\) 

\begin{question}Solve $\log_{ 4 }(x + 6) + \log_{ 4 }(x + 66) = 4$
    \soln{10cm}{Using the product rule for logarithms gives $\log_{ 4 }(\left(x + 6\right) \left(x + 66\right)) $ and rewriting in exponential form gives $\left(x + 6\right) \left(x + 66\right) = 256$ expanding and setting the equation equal to zero gives $x^{2} + 72 x + 140 = 0$.  Factoring gives $\left(x + 2\right) \left(x + 70\right)=0$. This gives two possible solutions $x=-70$ or $x=-2$. $x=-70$ is an extraneous solution.  The only soution is $x=-2$. }

\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_{ 4 }(x + 6) + \log_{ 4 }(x + 66) = 4 " src="/equation_images/%20%5Cdisplaystyle%20%5Clog_%7B%204%20%7D%28x%20%2B%206%29%20%2B%20%5Clog_%7B%204%20%7D%28x%20%2B%2066%29%20%3D%204%20" alt="LaTeX:  \displaystyle \log_{ 4 }(x + 6) + \log_{ 4 }(x + 66) = 4 " data-equation-content=" \displaystyle \log_{ 4 }(x + 6) + \log_{ 4 }(x + 66) = 4 " /> </p> </p>
HTML for Canvas
<p> <p>Using the product rule for logarithms gives  <img class="equation_image" title=" \displaystyle \log_{ 4 }(\left(x + 6\right) \left(x + 66\right))  " src="/equation_images/%20%5Cdisplaystyle%20%5Clog_%7B%204%20%7D%28%5Cleft%28x%20%2B%206%5Cright%29%20%5Cleft%28x%20%2B%2066%5Cright%29%29%20%20" alt="LaTeX:  \displaystyle \log_{ 4 }(\left(x + 6\right) \left(x + 66\right))  " data-equation-content=" \displaystyle \log_{ 4 }(\left(x + 6\right) \left(x + 66\right))  " />  and rewriting in exponential form gives  <img class="equation_image" title=" \displaystyle \left(x + 6\right) \left(x + 66\right) = 256 " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28x%20%2B%206%5Cright%29%20%5Cleft%28x%20%2B%2066%5Cright%29%20%3D%20256%20" alt="LaTeX:  \displaystyle \left(x + 6\right) \left(x + 66\right) = 256 " data-equation-content=" \displaystyle \left(x + 6\right) \left(x + 66\right) = 256 " />  expanding and setting the equation equal to zero gives  <img class="equation_image" title=" \displaystyle x^{2} + 72 x + 140 = 0 " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%20%2B%2072%20x%20%2B%20140%20%3D%200%20" alt="LaTeX:  \displaystyle x^{2} + 72 x + 140 = 0 " data-equation-content=" \displaystyle x^{2} + 72 x + 140 = 0 " /> .  Factoring gives  <img class="equation_image" title=" \displaystyle \left(x + 2\right) \left(x + 70\right)=0 " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28x%20%2B%202%5Cright%29%20%5Cleft%28x%20%2B%2070%5Cright%29%3D0%20" alt="LaTeX:  \displaystyle \left(x + 2\right) \left(x + 70\right)=0 " data-equation-content=" \displaystyle \left(x + 2\right) \left(x + 70\right)=0 " /> . This gives two possible solutions  <img class="equation_image" title=" \displaystyle x=-70 " src="/equation_images/%20%5Cdisplaystyle%20x%3D-70%20" alt="LaTeX:  \displaystyle x=-70 " data-equation-content=" \displaystyle x=-70 " />  or  <img class="equation_image" title=" \displaystyle x=-2 " src="/equation_images/%20%5Cdisplaystyle%20x%3D-2%20" alt="LaTeX:  \displaystyle x=-2 " data-equation-content=" \displaystyle x=-2 " /> .  <img class="equation_image" title=" \displaystyle x=-70 " src="/equation_images/%20%5Cdisplaystyle%20x%3D-70%20" alt="LaTeX:  \displaystyle x=-70 " data-equation-content=" \displaystyle x=-70 " />  is an extraneous solution.  The only soution is  <img class="equation_image" title=" \displaystyle x=-2 " src="/equation_images/%20%5Cdisplaystyle%20x%3D-2%20" alt="LaTeX:  \displaystyle x=-2 " data-equation-content=" \displaystyle x=-2 " /> . </p> </p>