Problem 73 - Introductory and Intermediate Algebra for College Students 4th

Question

Let \(\log _{b} 2=A\) and \(\log _{b} 3=C .\) Write each expression in terms of \(A\) and \(C\). $$\log _{b} \sqrt{\frac{2}{27}}$$

Step-by-Step Solution

Answer

The expression \(\log _{b} \sqrt{\frac{2}{27}}\) converted in terms of \(A\) and \(C\) is \(0.5 ( A - 3C )\)

Step 1 - Represent Square Roots as Fractional Exponent

Rewrite \(\sqrt{\frac{2}{27}}\) as \(\left(\frac{2}{27}\right)^{0.5}\). So, the expression becomes: \(\log _{b} \left(\frac{2}{27}\right)^{0.5}\)

Step 2 - Apply the Power Rule of Logarithms

The power rule of logarithms states that \(\log_b M^n=n \log_b M\). Accordingly, the expression from step 1 becomes: \(0.5 \log _{b} \left(\frac{2}{27}\right)\)

Step 3 - Apply Logarithm Rule for Fractions

Using the rule \(\log_b \frac{M}{N} = \log_b M - \log_b N\), the quantity within the log from step 2 can be separated: \(0.5 \left( \log _{b} 2 - \log _{b} 27 \right)\)

Step 4 - Break Down 27 into Base 3

Since we want to write the expression in terms of \(\log _{b} 2\) and \(\log _{b} 3\), rewrite 27 as \(3^3\). Now, the expression becomes: \(0.5 \left( \log _{b} 2 - \log _{b} (3^3) \right)\)

Step 5 - Apply Power Rule & Write in terms of A and C

Using the power rule of logarithms to the expression \(\log _{b} (3^3)\), it comes out to be \(3 \log _{b} 3\). Substituting \(\log _{b} 2 = A\) and \(\log _{b} 3 = C\), the final expression will be: \(0.5 ( A - 3C )\)