Question
Solve the given equation for \( x \) \( 2^{\frac{x}{2}}=5 \) \( x= \)
Ask by Franklin Lyons. in the United States
Mar 13,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x = 2 \log_{2}(5) \)
Solution
Solve the equation \( 2^{x/2}=5 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(2^{\frac{x}{2}}=5\)
- step1: Take the logarithm of both sides:
\(\log_{2}{\left(2^{\frac{x}{2}}\right)}=\log_{2}{\left(5\right)}\)
- step2: Evaluate the logarithm:
\(\frac{x}{2}=\log_{2}{\left(5\right)}\)
- step3: Cross multiply:
\(x=2\log_{2}{\left(5\right)}\)
The solution to the equation \(2^{\frac{x}{2}}=5\) is \(x=2\log_{2}(5)\).
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Simplify this solution Mind Expander
To solve the equation \( 2^{\frac{x}{2}} = 5 \), you can take the logarithm of both sides. Applying the logarithm, you get: \[ \log(2^{\frac{x}{2}}) = \log(5) \] Using the power rule of logarithms, this simplifies to: \[ \frac{x}{2} \cdot \log(2) = \log(5) \] Now, solving for \( x \): \[ x = 2 \cdot \frac{\log(5)}{\log(2)} \] Calculating the logs and substituting their values, you can determine the numerical value of \( x \). Plugging it into a calculator, you will find: \[ x \approx 4.644 \]
