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Questions: Algebra BusinessCalculus
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Solve the equation \(\displaystyle \log_{3}(x + 37)-\log_{3}(x + 19)=1\).
Using the quotient property of logarithms gives \(\displaystyle \log_{3}\frac{x + 37}{x + 19} = 1\). Making both sides of the equation exponents on the base \(\displaystyle 3\) gives \(\displaystyle \frac{x + 37}{x + 19}=3\). Clearing the fractions by multiplying by the LCD gives \(\displaystyle x + 37=3 x + 57\). Isolating \(\displaystyle x\) gives \(\displaystyle x = -10\).
\begin{question}Solve the equation $\log_{3}(x + 37)-\log_{3}(x + 19)=1$.
\soln{9cm}{Using the quotient property of logarithms gives $\log_{3}\frac{x + 37}{x + 19} = 1$. Making both sides of the equation exponents on the base $3$ gives $\frac{x + 37}{x + 19}=3$. Clearing the fractions by multiplying by the LCD gives $x + 37=3 x + 57$. Isolating $x$ gives $x = -10$. }
\end{question}
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\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Solve the equation <img class="equation_image" title=" \displaystyle \log_{3}(x + 37)-\log_{3}(x + 19)=1 " src="/equation_images/%20%5Cdisplaystyle%20%5Clog_%7B3%7D%28x%20%2B%2037%29-%5Clog_%7B3%7D%28x%20%2B%2019%29%3D1%20" alt="LaTeX: \displaystyle \log_{3}(x + 37)-\log_{3}(x + 19)=1 " data-equation-content=" \displaystyle \log_{3}(x + 37)-\log_{3}(x + 19)=1 " /> . </p> </p><p> <p>Using the quotient property of logarithms gives <img class="equation_image" title=" \displaystyle \log_{3}\frac{x + 37}{x + 19} = 1 " src="/equation_images/%20%5Cdisplaystyle%20%5Clog_%7B3%7D%5Cfrac%7Bx%20%2B%2037%7D%7Bx%20%2B%2019%7D%20%3D%201%20" alt="LaTeX: \displaystyle \log_{3}\frac{x + 37}{x + 19} = 1 " data-equation-content=" \displaystyle \log_{3}\frac{x + 37}{x + 19} = 1 " /> . Making both sides of the equation exponents on the base <img class="equation_image" title=" \displaystyle 3 " src="/equation_images/%20%5Cdisplaystyle%203%20" alt="LaTeX: \displaystyle 3 " data-equation-content=" \displaystyle 3 " /> gives <img class="equation_image" title=" \displaystyle \frac{x + 37}{x + 19}=3 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7Bx%20%2B%2037%7D%7Bx%20%2B%2019%7D%3D3%20" alt="LaTeX: \displaystyle \frac{x + 37}{x + 19}=3 " data-equation-content=" \displaystyle \frac{x + 37}{x + 19}=3 " /> . Clearing the fractions by multiplying by the LCD gives <img class="equation_image" title=" \displaystyle x + 37=3 x + 57 " src="/equation_images/%20%5Cdisplaystyle%20x%20%2B%2037%3D3%20x%20%2B%2057%20" alt="LaTeX: \displaystyle x + 37=3 x + 57 " data-equation-content=" \displaystyle x + 37=3 x + 57 " /> . 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 = -10 " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%20-10%20" alt="LaTeX: \displaystyle x = -10 " data-equation-content=" \displaystyle x = -10 " /> . </p> </p>