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Algebra
Logarithms
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Solve the equation \(\displaystyle \log_{8}(x + 32753)-\log_{8}(x + 49)=3\).

Using the quotient property of logarithms gives \(\displaystyle \log_{8}\frac{x + 32753}{x + 49} = 3\). Making both sides of the equation exponents on the base \(\displaystyle 8\) gives \(\displaystyle \frac{x + 32753}{x + 49}=512\). Clearing the fractions by multiplying by the LCD gives \(\displaystyle x + 32753=512 x + 25088\). Isolating \(\displaystyle x\) gives \(\displaystyle x = 15\).

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\begin{question}Solve the equation $\log_{8}(x + 32753)-\log_{8}(x + 49)=3$. 
    \soln{9cm}{Using the quotient property of logarithms gives $\log_{8}\frac{x + 32753}{x + 49} = 3$. Making both sides of the equation exponents on the base $8$ gives $\frac{x + 32753}{x + 49}=512$. Clearing the fractions by multiplying by the LCD gives $x + 32753=512 x + 25088$. Isolating $x$ gives $x = 15$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Solve the equation  <img class="equation_image" title=" \displaystyle \log_{8}(x + 32753)-\log_{8}(x + 49)=3 " src="/equation_images/%20%5Cdisplaystyle%20%5Clog_%7B8%7D%28x%20%2B%2032753%29-%5Clog_%7B8%7D%28x%20%2B%2049%29%3D3%20" alt="LaTeX:  \displaystyle \log_{8}(x + 32753)-\log_{8}(x + 49)=3 " data-equation-content=" \displaystyle \log_{8}(x + 32753)-\log_{8}(x + 49)=3 " /> . </p> </p>
HTML for Canvas
<p> <p>Using the quotient property of logarithms gives  <img class="equation_image" title=" \displaystyle \log_{8}\frac{x + 32753}{x + 49} = 3 " src="/equation_images/%20%5Cdisplaystyle%20%5Clog_%7B8%7D%5Cfrac%7Bx%20%2B%2032753%7D%7Bx%20%2B%2049%7D%20%3D%203%20" alt="LaTeX:  \displaystyle \log_{8}\frac{x + 32753}{x + 49} = 3 " data-equation-content=" \displaystyle \log_{8}\frac{x + 32753}{x + 49} = 3 " /> . Making both sides of the equation exponents on the base  <img class="equation_image" title=" \displaystyle 8 " src="/equation_images/%20%5Cdisplaystyle%208%20" alt="LaTeX:  \displaystyle 8 " data-equation-content=" \displaystyle 8 " />  gives  <img class="equation_image" title=" \displaystyle \frac{x + 32753}{x + 49}=512 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7Bx%20%2B%2032753%7D%7Bx%20%2B%2049%7D%3D512%20" alt="LaTeX:  \displaystyle \frac{x + 32753}{x + 49}=512 " data-equation-content=" \displaystyle \frac{x + 32753}{x + 49}=512 " /> . Clearing the fractions by multiplying by the LCD gives  <img class="equation_image" title=" \displaystyle x + 32753=512 x + 25088 " src="/equation_images/%20%5Cdisplaystyle%20x%20%2B%2032753%3D512%20x%20%2B%2025088%20" alt="LaTeX:  \displaystyle x + 32753=512 x + 25088 " data-equation-content=" \displaystyle x + 32753=512 x + 25088 " /> . Isolating  <img class="equation_image" title=" \displaystyle x " src="/equation_images/%20%5Cdisplaystyle%20x%20" alt="LaTeX:  \displaystyle x " data-equation-content=" \displaystyle x " />  gives  <img class="equation_image" title=" \displaystyle x = 15 " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%2015%20" alt="LaTeX:  \displaystyle x = 15 " data-equation-content=" \displaystyle x = 15 " /> . </p> </p>