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Solve \(\displaystyle \log_{8}(x + 10)+\log_{8}(x + 250) = 4\).

Using logarithmic properties and expanding the argument gives \(\displaystyle \log_{8}(x^{2} + 260 x + 2500)=4\). Making both sides an exponent on the base gives \(\displaystyle x^{2} + 260 x + 2500=8^{4}\). Expanding and setting equal to zero gives \(\displaystyle x^{2} + 260 x - 1596=0\). Factoring gives \(\displaystyle \left(x - 6\right) \left(x + 266\right)=0\). Solving gives the two possible solutions \(\displaystyle x = -266\) and \(\displaystyle x = 6\). The domain of the original is \(\displaystyle \left(-10, \infty\right) \bigcap \left(-250, \infty\right)=\left(-10, \infty\right)\). Checking if each possible solution is in the domain gives: \(\displaystyle x = -266\) is not a solution. \(\displaystyle x=6\) is a solution.

Download \(\LaTeX\) 

\begin{question}Solve $\log_{8}(x + 10)+\log_{8}(x + 250) = 4$. 
    \soln{9cm}{Using logarithmic properties and expanding the argument gives $\log_{8}(x^{2} + 260 x + 2500)=4$. Making both sides an exponent on the base gives $x^{2} + 260 x + 2500=8^{4}$. Expanding and setting equal to zero gives $x^{2} + 260 x - 1596=0$. Factoring gives $\left(x - 6\right) \left(x + 266\right)=0$. Solving gives the two possible solutions $x = -266$ and $x = 6$. The domain of the original is $\left(-10, \infty\right) \bigcap \left(-250, \infty\right)=\left(-10, \infty\right)$.  Checking if each possible solution is in the domain gives: $x = -266$ is not a solution. $x=6$ 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 + 10)+\log_{8}(x + 250) = 4 " src="/equation_images/%20%5Cdisplaystyle%20%5Clog_%7B8%7D%28x%20%2B%2010%29%2B%5Clog_%7B8%7D%28x%20%2B%20250%29%20%3D%204%20" alt="LaTeX:  \displaystyle \log_{8}(x + 10)+\log_{8}(x + 250) = 4 " data-equation-content=" \displaystyle \log_{8}(x + 10)+\log_{8}(x + 250) = 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} + 260 x + 2500)=4 " src="/equation_images/%20%5Cdisplaystyle%20%5Clog_%7B8%7D%28x%5E%7B2%7D%20%2B%20260%20x%20%2B%202500%29%3D4%20" alt="LaTeX:  \displaystyle \log_{8}(x^{2} + 260 x + 2500)=4 " data-equation-content=" \displaystyle \log_{8}(x^{2} + 260 x + 2500)=4 " /> . Making both sides an exponent on the base gives  <img class="equation_image" title=" \displaystyle x^{2} + 260 x + 2500=8^{4} " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%20%2B%20260%20x%20%2B%202500%3D8%5E%7B4%7D%20" alt="LaTeX:  \displaystyle x^{2} + 260 x + 2500=8^{4} " data-equation-content=" \displaystyle x^{2} + 260 x + 2500=8^{4} " /> . Expanding and setting equal to zero gives  <img class="equation_image" title=" \displaystyle x^{2} + 260 x - 1596=0 " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%20%2B%20260%20x%20-%201596%3D0%20" alt="LaTeX:  \displaystyle x^{2} + 260 x - 1596=0 " data-equation-content=" \displaystyle x^{2} + 260 x - 1596=0 " /> . Factoring gives  <img class="equation_image" title=" \displaystyle \left(x - 6\right) \left(x + 266\right)=0 " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28x%20-%206%5Cright%29%20%5Cleft%28x%20%2B%20266%5Cright%29%3D0%20" alt="LaTeX:  \displaystyle \left(x - 6\right) \left(x + 266\right)=0 " data-equation-content=" \displaystyle \left(x - 6\right) \left(x + 266\right)=0 " /> . Solving gives the two possible solutions  <img class="equation_image" title=" \displaystyle x = -266 " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%20-266%20" alt="LaTeX:  \displaystyle x = -266 " data-equation-content=" \displaystyle x = -266 " />  and  <img class="equation_image" title=" \displaystyle x = 6 " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%206%20" alt="LaTeX:  \displaystyle x = 6 " data-equation-content=" \displaystyle x = 6 " /> . The domain of the original is  <img class="equation_image" title=" \displaystyle \left(-10, \infty\right) \bigcap \left(-250, \infty\right)=\left(-10, \infty\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28-10%2C%20%5Cinfty%5Cright%29%20%5Cbigcap%20%5Cleft%28-250%2C%20%5Cinfty%5Cright%29%3D%5Cleft%28-10%2C%20%5Cinfty%5Cright%29%20" alt="LaTeX:  \displaystyle \left(-10, \infty\right) \bigcap \left(-250, \infty\right)=\left(-10, \infty\right) " data-equation-content=" \displaystyle \left(-10, \infty\right) \bigcap \left(-250, \infty\right)=\left(-10, \infty\right) " /> .  Checking if each possible solution is in the domain gives:  <img class="equation_image" title=" \displaystyle x = -266 " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%20-266%20" alt="LaTeX:  \displaystyle x = -266 " data-equation-content=" \displaystyle x = -266 " />  is not a solution.  <img class="equation_image" title=" \displaystyle x=6 " src="/equation_images/%20%5Cdisplaystyle%20x%3D6%20" alt="LaTeX:  \displaystyle x=6 " data-equation-content=" \displaystyle x=6 " />  is a solution. </p> </p>