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Solve \(\displaystyle \log_{8}(x + 247)+\log_{8}(x + 7) = 4\).

Using logarithmic properties and expanding the argument gives \(\displaystyle \log_{8}(x^{2} + 254 x + 1729)=4\). Making both sides an exponent on the base gives \(\displaystyle x^{2} + 254 x + 1729=8^{4}\). Expanding and setting equal to zero gives \(\displaystyle x^{2} + 254 x - 2367=0\). Factoring gives \(\displaystyle \left(x - 9\right) \left(x + 263\right)=0\). Solving gives the two possible solutions \(\displaystyle x = -263\) and \(\displaystyle x = 9\). The domain of the original is \(\displaystyle \left(-247, \infty\right) \bigcap \left(-7, \infty\right)=\left(-7, \infty\right)\). Checking if each possible solution is in the domain gives: \(\displaystyle x = -263\) is not a solution. \(\displaystyle x=9\) is a solution.

Download \(\LaTeX\) 

\begin{question}Solve $\log_{8}(x + 247)+\log_{8}(x + 7) = 4$. 
    \soln{9cm}{Using logarithmic properties and expanding the argument gives $\log_{8}(x^{2} + 254 x + 1729)=4$. Making both sides an exponent on the base gives $x^{2} + 254 x + 1729=8^{4}$. Expanding and setting equal to zero gives $x^{2} + 254 x - 2367=0$. Factoring gives $\left(x - 9\right) \left(x + 263\right)=0$. Solving gives the two possible solutions $x = -263$ and $x = 9$. The domain of the original is $\left(-247, \infty\right) \bigcap \left(-7, \infty\right)=\left(-7, \infty\right)$.  Checking if each possible solution is in the domain gives: $x = -263$ is not a solution. $x=9$ is a solution. }

\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 + 247)+\log_{8}(x + 7) = 4 " src="/equation_images/%20%5Cdisplaystyle%20%5Clog_%7B8%7D%28x%20%2B%20247%29%2B%5Clog_%7B8%7D%28x%20%2B%207%29%20%3D%204%20" alt="LaTeX:  \displaystyle \log_{8}(x + 247)+\log_{8}(x + 7) = 4 " data-equation-content=" \displaystyle \log_{8}(x + 247)+\log_{8}(x + 7) = 4 " /> . </p> </p>
HTML for Canvas
<p> <p>Using logarithmic properties and expanding the argument gives  <img class="equation_image" title=" \displaystyle \log_{8}(x^{2} + 254 x + 1729)=4 " src="/equation_images/%20%5Cdisplaystyle%20%5Clog_%7B8%7D%28x%5E%7B2%7D%20%2B%20254%20x%20%2B%201729%29%3D4%20" alt="LaTeX:  \displaystyle \log_{8}(x^{2} + 254 x + 1729)=4 " data-equation-content=" \displaystyle \log_{8}(x^{2} + 254 x + 1729)=4 " /> . Making both sides an exponent on the base gives  <img class="equation_image" title=" \displaystyle x^{2} + 254 x + 1729=8^{4} " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%20%2B%20254%20x%20%2B%201729%3D8%5E%7B4%7D%20" alt="LaTeX:  \displaystyle x^{2} + 254 x + 1729=8^{4} " data-equation-content=" \displaystyle x^{2} + 254 x + 1729=8^{4} " /> . Expanding and setting equal to zero gives  <img class="equation_image" title=" \displaystyle x^{2} + 254 x - 2367=0 " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%20%2B%20254%20x%20-%202367%3D0%20" alt="LaTeX:  \displaystyle x^{2} + 254 x - 2367=0 " data-equation-content=" \displaystyle x^{2} + 254 x - 2367=0 " /> . Factoring gives  <img class="equation_image" title=" \displaystyle \left(x - 9\right) \left(x + 263\right)=0 " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28x%20-%209%5Cright%29%20%5Cleft%28x%20%2B%20263%5Cright%29%3D0%20" alt="LaTeX:  \displaystyle \left(x - 9\right) \left(x + 263\right)=0 " data-equation-content=" \displaystyle \left(x - 9\right) \left(x + 263\right)=0 " /> . Solving gives the two possible solutions  <img class="equation_image" title=" \displaystyle x = -263 " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%20-263%20" alt="LaTeX:  \displaystyle x = -263 " data-equation-content=" \displaystyle x = -263 " />  and  <img class="equation_image" title=" \displaystyle x = 9 " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%209%20" alt="LaTeX:  \displaystyle x = 9 " data-equation-content=" \displaystyle x = 9 " /> . The domain of the original is  <img class="equation_image" title=" \displaystyle \left(-247, \infty\right) \bigcap \left(-7, \infty\right)=\left(-7, \infty\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28-247%2C%20%5Cinfty%5Cright%29%20%5Cbigcap%20%5Cleft%28-7%2C%20%5Cinfty%5Cright%29%3D%5Cleft%28-7%2C%20%5Cinfty%5Cright%29%20" alt="LaTeX:  \displaystyle \left(-247, \infty\right) \bigcap \left(-7, \infty\right)=\left(-7, \infty\right) " data-equation-content=" \displaystyle \left(-247, \infty\right) \bigcap \left(-7, \infty\right)=\left(-7, \infty\right) " /> .  Checking if each possible solution is in the domain gives:  <img class="equation_image" title=" \displaystyle x = -263 " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%20-263%20" alt="LaTeX:  \displaystyle x = -263 " data-equation-content=" \displaystyle x = -263 " />  is not a solution.  <img class="equation_image" title=" \displaystyle x=9 " src="/equation_images/%20%5Cdisplaystyle%20x%3D9%20" alt="LaTeX:  \displaystyle x=9 " data-equation-content=" \displaystyle x=9 " />  is a solution. </p> </p>