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Questions: Algebra BusinessCalculus
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Solve the equation \(\displaystyle \log_{3}(x + 29)-\log_{3}(x + 11)=1\).
Using the quotient property of logarithms gives \(\displaystyle \log_{3}\frac{x + 29}{x + 11} = 1\). Making both sides of the equation exponents on the base \(\displaystyle 3\) gives \(\displaystyle \frac{x + 29}{x + 11}=3\). Clearing the fractions by multiplying by the LCD gives \(\displaystyle x + 29=3 x + 33\). Isolating \(\displaystyle x\) gives \(\displaystyle x = -2\).
\begin{question}Solve the equation $\log_{3}(x + 29)-\log_{3}(x + 11)=1$.
\soln{9cm}{Using the quotient property of logarithms gives $\log_{3}\frac{x + 29}{x + 11} = 1$. Making both sides of the equation exponents on the base $3$ gives $\frac{x + 29}{x + 11}=3$. Clearing the fractions by multiplying by the LCD gives $x + 29=3 x + 33$. Isolating $x$ gives $x = -2$. }
\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 + 29)-\log_{3}(x + 11)=1 " src="/equation_images/%20%5Cdisplaystyle%20%5Clog_%7B3%7D%28x%20%2B%2029%29-%5Clog_%7B3%7D%28x%20%2B%2011%29%3D1%20" alt="LaTeX: \displaystyle \log_{3}(x + 29)-\log_{3}(x + 11)=1 " data-equation-content=" \displaystyle \log_{3}(x + 29)-\log_{3}(x + 11)=1 " /> . </p> </p><p> <p>Using the quotient property of logarithms gives <img class="equation_image" title=" \displaystyle \log_{3}\frac{x + 29}{x + 11} = 1 " src="/equation_images/%20%5Cdisplaystyle%20%5Clog_%7B3%7D%5Cfrac%7Bx%20%2B%2029%7D%7Bx%20%2B%2011%7D%20%3D%201%20" alt="LaTeX: \displaystyle \log_{3}\frac{x + 29}{x + 11} = 1 " data-equation-content=" \displaystyle \log_{3}\frac{x + 29}{x + 11} = 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 + 29}{x + 11}=3 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7Bx%20%2B%2029%7D%7Bx%20%2B%2011%7D%3D3%20" alt="LaTeX: \displaystyle \frac{x + 29}{x + 11}=3 " data-equation-content=" \displaystyle \frac{x + 29}{x + 11}=3 " /> . Clearing the fractions by multiplying by the LCD gives <img class="equation_image" title=" \displaystyle x + 29=3 x + 33 " src="/equation_images/%20%5Cdisplaystyle%20x%20%2B%2029%3D3%20x%20%2B%2033%20" alt="LaTeX: \displaystyle x + 29=3 x + 33 " data-equation-content=" \displaystyle x + 29=3 x + 33 " /> . 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 = -2 " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%20-2%20" alt="LaTeX: \displaystyle x = -2 " data-equation-content=" \displaystyle x = -2 " /> . </p> </p>