If $20 = 3^x$, which of the following expresses $x$ as a base-ten logarithm?

Neetesh Kumar | October 23, 2024

College Algebra Homework Help

CLEPS College Algebra Guide Question 75
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Step-by-step solution:

We are given the equation:

$20 = 3^x$

To express ( x ) as a logarithm, we can take the logarithm of both sides of the equation. We use the base-10 logarithm (logarithm base 10) as specified in the question.

Step 1: Take the logarithm of both sides

Take the logarithm of both sides:

$\log_{10} 20 = \log_{10} (3^x)$

Step 2: Apply the logarithmic power rule

Use the power rule of logarithms, which states that:

$\log_b (a^c) = c \log_b a$

Applying this to ( \log_{10} (3^x) ), we get:

$\log_{10} 20 = x \log_{10} 3$

Step 3: Solve for ( x )

To isolate ( x ), divide both sides by ( \log_{10} 3 ):

$x = \frac{\log_{10} 20}{\log_{10} 3}$

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Final Answer:

The expression for ( x ) is:

$\text{(E) } \frac{\log_{10} 20}{\log_{10} 3}$

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