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Chapter 13: Problem 44

Write as the sum or difference of logarithms and simplify, if possible. Assume all variables represent positive real numbers. $$\log \frac{3}{a b^{2}}$$

Short Answer

Expert verified
The simplified expression is \( \log \frac{3}{ab^2} = \log 3 - \log a - 2 \log b \).

Step by step solution

01

Apply the Quotient Rule

To begin, we will apply the quotient rule for logarithms to rewrite the expression as a difference of logarithms: \[ \log \frac{3}{ab^2} = \log 3 - \log(ab^2) \] #Step 2: Apply the Product Rule#
02

Apply the Product Rule

Now, apply the product rule for logarithms, which states that \(\log(ab) = \log a + \log b\), to further break down the expression: \[ \log 3 - \log(ab^2) = \log 3 - (\log a + \log b^2) \] #Step 3: Simplify the Expression#
03

Simplify the Expression

Next, simplify the expression by distributing the negative sign to the terms inside the parenthesis: \[ \log 3 - (\log a + \log b^2) = \log 3 - \log a - \log b^2 \] #Step 4: Apply the Power Rule#
04

Apply the Power Rule

Finally, apply the power rule for logarithms to the last term in the expression: \[ \log 3 - \log a - \log b^2 = \log 3 - \log a - 2 \log b \] The simplified expression is \[ \log \frac{3}{ab^2} = \log 3 - \log a - 2 \log b \]

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