Question
solve for \( x \). Leave answer in radical form. \( \log _{4} x+\log _{4} 2 x=3 \)
Ask by Santiago Pearson. in South Africa
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x=4\sqrt{2} \)
Solution
Solve the equation \( \log_{4}x+\log_{4}2x=3 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(\log_{4}{\left(x\right)}+\log_{4}{\left(2x\right)}=3\)
- step1: Find the domain:
\(\log_{4}{\left(x\right)}+\log_{4}{\left(2x\right)}=3,x>0\)
- step2: Add the terms:
\(\log_{4}{\left(2x^{2}\right)}=3\)
- step3: Convert the logarithm into exponential form:
\(2x^{2}=4^{3}\)
- step4: Evaluate the power:
\(2x^{2}=64\)
- step5: Divide both sides:
\(\frac{2x^{2}}{2}=\frac{64}{2}\)
- step6: Divide the numbers:
\(x^{2}=32\)
- step7: Simplify the expression:
\(x=\pm \sqrt{32}\)
- step8: Simplify:
\(x=\pm 4\sqrt{2}\)
- step9: Separate into possible cases:
\(\begin{align}&x=4\sqrt{2}\\&x=-4\sqrt{2}\end{align}\)
- step10: Check if the solution is in the defined range:
\(\begin{align}&x=4\sqrt{2}\\&x=-4\sqrt{2}\end{align},x>0\)
- step11: Find the intersection:
\(x=4\sqrt{2}\)
The solution to the equation \( \log_{4}x+\log_{4}2x=3 \) is \( x=4\sqrt{2} \).
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Simplify this solution Extra Insights
To solve the equation \( \log _{4} x+\log _{4} 2x=3 \), we can use the property of logarithms that allows us to combine them: \[ \log_{4} x + \log_{4} (2x) = \log_{4} (x \cdot 2x) = \log_{4} (2x^2) \] Thus, the equation becomes: \[ \log_{4} (2x^2) = 3 \] Now, we can rewrite this in exponential form: \[ 2x^2 = 4^3 \] Calculating \( 4^3 \): \[ 2x^2 = 64 \] Now, divide both sides by 2: \[ x^2 = \frac{64}{2} = 32 \] Taking the square root of both sides, we get: \[ x = \sqrt{32} \] Noting that \( \sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2} \). In radical form, the solution is: \[ x = 4\sqrt{2} \]
