Question
Solve for \( x \), rounding to the nearest hundredth. \[ 201 \cdot 2^{x}=67 \]
Ask by Bartlett Murphy. in the United States
Jan 28,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x \approx -1.58 \)
Solution
Solve the equation \( 201 \cdot 2^{x}=67 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(201\times 2^{x}=67\)
- step1: Divide both sides:
\(\frac{201\times 2^{x}}{201}=\frac{67}{201}\)
- step2: Divide the numbers:
\(2^{x}=\frac{1}{3}\)
- step3: Take the logarithm of both sides:
\(\log_{2}{\left(2^{x}\right)}=\log_{2}{\left(\frac{1}{3}\right)}\)
- step4: Evaluate the logarithm:
\(x=\log_{2}{\left(\frac{1}{3}\right)}\)
- step5: Simplify:
\(x=-\log_{2}{\left(3\right)}\)
The solution to the equation \(201 \cdot 2^{x}=67\) is \(x = -\log_{2}{(3)}\).
To round this value to the nearest hundredth, we can calculate it as follows:
\[ x = -\log_{2}{(3)} \approx -1.58496 \]
Rounded to the nearest hundredth, the value of \(x\) is approximately -1.58.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To solve for \( x \) in the equation \( 201 \cdot 2^{x}=67 \), we first isolate \( 2^x \): \[ 2^x = \frac{67}{201} \] Next, we calculate \( \frac{67}{201} \): \[ \frac{67}{201} \approx 0.3333 \] Now, we take the logarithm of both sides to solve for \( x \): \[ \log(2^x) = \log(0.3333) \] Using the power rule of logarithms, we have: \[ x \cdot \log(2) = \log(0.3333) \] Now, we can solve for \( x \): \[ x = \frac{\log(0.3333)}{\log(2)} \] Calculating: \[ \log(0.3333) \approx -0.4771 \] \[ \log(2) \approx 0.3010 \] Thus, \[ x \approx \frac{-0.4771}{0.3010} \approx -1.585 \] Rounding to the nearest hundredth, we get: \[ \boxed{-1.59} \]
