Question
Solve for \( x \) : \( 4^{x}=38 \) \( x=\square \) You may enter the exact value or round to 4 decimal places. Question Help: Video
Ask by Willis Lynch. in the United States
Mar 13,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x \approx 2.6240 \)
Solution
Solve the equation \( 4^{x}=38 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(4^{x}=38\)
- step1: Take the logarithm of both sides:
\(\log_{4}{\left(4^{x}\right)}=\log_{4}{\left(38\right)}\)
- step2: Evaluate the logarithm:
\(x=\log_{4}{\left(38\right)}\)
- step3: Simplify:
\(x=\frac{\log_{2}{\left(38\right)}}{2}\)
- step4: Calculate:
\(x=\frac{1+\log_{2}{\left(19\right)}}{2}\)
The solution to the equation \(4^{x}=38\) is \(x \approx 2.623964\).
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Simplify this solution Bonus Knowledge
To solve for \( x \) in the equation \( 4^{x}=38 \), you can take the logarithm of both sides. Using the base 10 logarithm, the equation becomes: \[ x \log(4) = \log(38) \] Now, isolate \( x \): \[ x = \frac{\log(38)}{\log(4)} \] Calculating the values gives you: \[ x \approx \frac{1.5798}{0.6021} \approx 2.6210 \] So, \( x \approx 2.6210 \) when rounded to four decimal places.
