Question

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where​ possible, evaluate logarithmic expressions.
log Subscript 3 Baseline 54 minus log Subscript 3 Baseline 2
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Part 1
log Subscript 3 Baseline 54 minus log Subscript 3 Baseline 2equals
  
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​(Type an exact answer in simplified form. Use integers or fractions for any numbers in the​ expression.)

Answer 1

To condense the logarithmic expression \( \log_3 54 - \log_3 2\), you can use the property of logarithms that states:

\[
\log_b a - \log_b c = \log_b \left( \frac{a}{c} \right)
\]

Applying this property to your expression:

\[
\log_3 54 - \log_3 2 = \log_3 \left( \frac{54}{2} \right)
\]

Now, simplify the fraction:

\[
\frac{54}{2} = 27
\]

Thus, the expression becomes:

\[
\log_3 \left( \frac{54}{2} \right) = \log_3 27
\]

So, we rewrite the original expression as:

\[
\log_3 54 - \log_3 2 = \log_3 27
\]

Now, since \(27\) can be expressed as \(3^3\), we can evaluate the logarithm:

\[
\log_3 27 = \log_3 (3^3) = 3
\]

Therefore, the condensed expression is:

\[
\log_3 54 - \log_3 2 = \log_3 27 = 3
\]

In summary, the final result is:

\[
\log_3 54 - \log_3 2 = \log_3 27 = 3
\]