Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( r = \frac{\log_{11}(323) - \log_{11}(90) - 6}{3} \)
Solution
Solve the equation \( -9 \cdot 11^{3r+6}=-32.3 \).
Solve the equation by following steps:
- step0: Solve for \(r\):
\(-9\times 11^{3r+6}=-32.3\)
- step1: Change the sign:
\(9\times 11^{3r+6}=32.3\)
- step2: Divide both sides:
\(\frac{9\times 11^{3r+6}}{9}=\frac{32.3}{9}\)
- step3: Divide the numbers:
\(11^{3r+6}=\frac{323}{90}\)
- step4: Take the logarithm of both sides:
\(\log_{11}{\left(11^{3r+6}\right)}=\log_{11}{\left(\frac{323}{90}\right)}\)
- step5: Evaluate the logarithm:
\(3r+6=\log_{11}{\left(\frac{323}{90}\right)}\)
- step6: Move the constant to the right side:
\(3r=\log_{11}{\left(\frac{323}{90}\right)}-6\)
- step7: Divide both sides:
\(\frac{3r}{3}=\frac{\log_{11}{\left(\frac{323}{90}\right)}-6}{3}\)
- step8: Divide the numbers:
\(r=\frac{\log_{11}{\left(\frac{323}{90}\right)}-6}{3}\)
- step9: Simplify:
\(r=\frac{\log_{11}{\left(323\right)}-\log_{11}{\left(90\right)}-6}{3}\)
The solution to the equation \( -9 \cdot 11^{3r+6}=-32.3 \) is \( r=\frac{\log_{11}{(323)}-\log_{11}{(90)}-6}{3} \).
Reviewed and approved by the UpStudy tutoring team
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Explain 
Simplify this solution Beyond the Answer
To solve the equation \( -9 \cdot 11^{3r+6} = -32.3 \), you can start by dividing both sides by -9 to simplify. This gives you \( 11^{3r+6} = \frac{32.3}{9} \). Then you'll want to calculate \( \frac{32.3}{9} \) to get approximately 3.5889. Next, you will apply logarithms to both sides to solve for \( r \). Taking the logarithm gives you \( 3r + 6 = \log_{11}(3.5889) \). After calculating \( \log_{11}(3.5889) \) (this may require change of base), you can isolate \( r \) to find the solution. Lastly, be careful with your calculations along the way, as small mistakes can lead to drastically different answers!
