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Algebra
Logarithms
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Solve \(\displaystyle \log_{ 9 }(x + 12) + \log_{ 9 }(x + 84) = 3\)

Using the product rule for logarithms gives \(\displaystyle \log_{ 9 }(\left(x + 12\right) \left(x + 84\right)) \) and rewriting in exponential form gives \(\displaystyle \left(x + 12\right) \left(x + 84\right) = 729\) expanding and setting the equation equal to zero gives \(\displaystyle x^{2} + 96 x + 279 = 0\). Factoring gives \(\displaystyle \left(x + 3\right) \left(x + 93\right)=0\). This gives two possible solutions \(\displaystyle x=-93\) or \(\displaystyle x=-3\). \(\displaystyle x=-93\) is an extraneous solution. The only soution is \(\displaystyle x=-3\).

Download \(\LaTeX\) 

\begin{question}Solve $\log_{ 9 }(x + 12) + \log_{ 9 }(x + 84) = 3$
    \soln{10cm}{Using the product rule for logarithms gives $\log_{ 9 }(\left(x + 12\right) \left(x + 84\right)) $ and rewriting in exponential form gives $\left(x + 12\right) \left(x + 84\right) = 729$ expanding and setting the equation equal to zero gives $x^{2} + 96 x + 279 = 0$.  Factoring gives $\left(x + 3\right) \left(x + 93\right)=0$. This gives two possible solutions $x=-93$ or $x=-3$. $x=-93$ is an extraneous solution.  The only soution is $x=-3$. }

\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_{ 9 }(x + 12) + \log_{ 9 }(x + 84) = 3 " src="/equation_images/%20%5Cdisplaystyle%20%5Clog_%7B%209%20%7D%28x%20%2B%2012%29%20%2B%20%5Clog_%7B%209%20%7D%28x%20%2B%2084%29%20%3D%203%20" alt="LaTeX:  \displaystyle \log_{ 9 }(x + 12) + \log_{ 9 }(x + 84) = 3 " data-equation-content=" \displaystyle \log_{ 9 }(x + 12) + \log_{ 9 }(x + 84) = 3 " /> </p> </p>
HTML for Canvas
<p> <p>Using the product rule for logarithms gives  <img class="equation_image" title=" \displaystyle \log_{ 9 }(\left(x + 12\right) \left(x + 84\right))  " src="/equation_images/%20%5Cdisplaystyle%20%5Clog_%7B%209%20%7D%28%5Cleft%28x%20%2B%2012%5Cright%29%20%5Cleft%28x%20%2B%2084%5Cright%29%29%20%20" alt="LaTeX:  \displaystyle \log_{ 9 }(\left(x + 12\right) \left(x + 84\right))  " data-equation-content=" \displaystyle \log_{ 9 }(\left(x + 12\right) \left(x + 84\right))  " />  and rewriting in exponential form gives  <img class="equation_image" title=" \displaystyle \left(x + 12\right) \left(x + 84\right) = 729 " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28x%20%2B%2012%5Cright%29%20%5Cleft%28x%20%2B%2084%5Cright%29%20%3D%20729%20" alt="LaTeX:  \displaystyle \left(x + 12\right) \left(x + 84\right) = 729 " data-equation-content=" \displaystyle \left(x + 12\right) \left(x + 84\right) = 729 " />  expanding and setting the equation equal to zero gives  <img class="equation_image" title=" \displaystyle x^{2} + 96 x + 279 = 0 " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%20%2B%2096%20x%20%2B%20279%20%3D%200%20" alt="LaTeX:  \displaystyle x^{2} + 96 x + 279 = 0 " data-equation-content=" \displaystyle x^{2} + 96 x + 279 = 0 " /> .  Factoring gives  <img class="equation_image" title=" \displaystyle \left(x + 3\right) \left(x + 93\right)=0 " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28x%20%2B%203%5Cright%29%20%5Cleft%28x%20%2B%2093%5Cright%29%3D0%20" alt="LaTeX:  \displaystyle \left(x + 3\right) \left(x + 93\right)=0 " data-equation-content=" \displaystyle \left(x + 3\right) \left(x + 93\right)=0 " /> . This gives two possible solutions  <img class="equation_image" title=" \displaystyle x=-93 " src="/equation_images/%20%5Cdisplaystyle%20x%3D-93%20" alt="LaTeX:  \displaystyle x=-93 " data-equation-content=" \displaystyle x=-93 " />  or  <img class="equation_image" title=" \displaystyle x=-3 " src="/equation_images/%20%5Cdisplaystyle%20x%3D-3%20" alt="LaTeX:  \displaystyle x=-3 " data-equation-content=" \displaystyle x=-3 " /> .  <img class="equation_image" title=" \displaystyle x=-93 " src="/equation_images/%20%5Cdisplaystyle%20x%3D-93%20" alt="LaTeX:  \displaystyle x=-93 " data-equation-content=" \displaystyle x=-93 " />  is an extraneous solution.  The only soution is  <img class="equation_image" title=" \displaystyle x=-3 " src="/equation_images/%20%5Cdisplaystyle%20x%3D-3%20" alt="LaTeX:  \displaystyle x=-3 " data-equation-content=" \displaystyle x=-3 " /> . </p> </p>