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Chapter 3: Problem 53

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$6 \log _{3}(0.5 x)=11$$

Short Answer

Expert verified
To solve the equation \(6 \log _{3}(0.5 x)=11\), the variable x calculates to be \(\frac{3^{\frac{11}{6}}}{0.5}\), which approximates to a numerical value, rounded off to 3 decimal places.

Step by step solution

01

Isolation of the Logarithmic Expression

You need to first isolate the logarithmic expression on one side of the equation. Start with the given equation \(6 \log_{3}(0.5x) = 11\) . To isolate the logarithm, divide both sides by 6. That gives: \(\log_{3}(0.5x) = \frac{11}{6}\)
02

Rewrite Equation in exponential form

Once the logarithmic equation is isolated, it can be rewritten in its exponential form. Using the base of the logarithm, 3, as the base of the exponent, the equation becomes: \(3^{\frac{11}{6}} = 0.5x\)
03

Solve for the variable x

Armed with the equation \(3^{\frac{11}{6}} = 0.5x\), you can then solve for the variable x by dividing both sides by 0.5. This gives \(x = \frac{3^{\frac{11}{6}}}{0.5}\)
04

Deriving Numerical Value for x

Lastly, evaluate the numerical value of x by calculating \(\frac{3^{\frac{11}{6}}}{0.5}\). After evaluating, round off to 3 decimal places to get the final answer

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