Harrington Marshall
07/31/2023 · Middle School
Solve: \( \log 2 x+\log 3=\log 30 \) \( \begin{array}{l}x=\frac{\log 30-\log 3}{\log 2} \\ x=6 \\ x=\frac{27}{2} \\ x=5\end{array} \)
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Tutor-Verified Answer
Step-by-step Solution
Solve the equation \( \log(2x)+\log(3)=\log(30) \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(\log_{10}{\left(2x\right)}+\log_{10}{\left(3\right)}=\log_{10}{\left(30\right)}\)
- step1: Find the domain:
\(\log_{10}{\left(2x\right)}+\log_{10}{\left(3\right)}=\log_{10}{\left(30\right)},x>0\)
- step2: Add the terms:
\(\log_{10}{\left(6x\right)}=\log_{10}{\left(30\right)}\)
- step3: Evaluate the logarithm:
\(6x=30\)
- step4: Divide both sides:
\(\frac{6x}{6}=\frac{30}{6}\)
- step5: Divide the numbers:
\(x=5\)
- step6: Check if the solution is in the defined range:
\(x=5,x>0\)
- step7: Find the intersection:
\(x=5\)
The solution to the equation \( \log(2x)+\log(3)=\log(30) \) is \( x=5 \).
Quick Answer
\( x=5 \)
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