Munoz Schneider
09/12/2024 · Primary School
Solve the logarithmic equation. Be sure to reject any value \[ \log (x+7)-\log 2=\log (4 x+1) \] Solve the equation to find the solution set. Select the corre
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Step-by-step Solution
Solve the equation \( \log (x+7)-\log 2=\log (4 x+1) \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(\log_{10}{\left(x+7\right)}-\log_{10}{\left(2\right)}=\log_{10}{\left(4x+1\right)}\)
- step1: Find the domain:
\(\log_{10}{\left(x+7\right)}-\log_{10}{\left(2\right)}=\log_{10}{\left(4x+1\right)},x>-\frac{1}{4}\)
- step2: Transform the expression:
\(\log_{10}{\left(\frac{x+7}{2}\right)}=\log_{10}{\left(4x+1\right)}\)
- step3: Set the arguments equal:
\(\frac{x+7}{2}=4x+1\)
- step4: Multiply both sides of the equation by LCD:
\(\frac{x+7}{2}\times 2=\left(4x+1\right)\times 2\)
- step5: Simplify the equation:
\(x+7=8x+2\)
- step6: Move the variable to the left side:
\(x+7-8x=2\)
- step7: Subtract the terms:
\(-7x+7=2\)
- step8: Move the constant to the right side:
\(-7x=2-7\)
- step9: Subtract the numbers:
\(-7x=-5\)
- step10: Change the signs:
\(7x=5\)
- step11: Divide both sides:
\(\frac{7x}{7}=\frac{5}{7}\)
- step12: Divide the numbers:
\(x=\frac{5}{7}\)
- step13: Check if the solution is in the defined range:
\(x=\frac{5}{7},x>-\frac{1}{4}\)
- step14: Find the intersection:
\(x=\frac{5}{7}\)
The solution to the logarithmic equation \(\log (x+7)-\log 2=\log (4x+1)\) is \(x=\frac{5}{7}\) or \(x=0.\dot{7}1428\dot{5}\).
Quick Answer
The solution to the equation is \(x=\frac{5}{7}\) or \(x=0.\dot{7}1428\dot{5}\).
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