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Chapter 4: Problem 22

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b}\left(x y^{3}\right) $$

Short Answer

Expert verified
The expanded form of the logarithmic expression \( \log_b(xy^3) \) is \( \log_b(x) + 3\log_b(y) \).

Step by step solution

01

Recall the Product Rule of Logarithms

Remember the product rule of logarithms which states that \( \log_b(xy) = \log_b(x) + \log_b(y) \). This rule allows us to separate a single logarithm of a product into a sum of multiple logarithms.
02

Apply the Product Rule

Applying this rule, we can rewrite the given logarithm \(\log_b(xy^3)\) as \(\log_b(x) + \log_b(y^3)\).
03

Recall the Power Rule of Logarithms

Now remember the power rule of logarithms that states \( \log_b (a^n) = n\log_b(a) \). This rule allows us to separate the exponent from the argument of the logarithm and bring it out front as a coefficient.
04

Apply the Power Rule

Applying this rule to our expression, we can rewrite the second term \( \log_b(y^3) \) as \( 3\log_b(y) \). So, the expanded form is \( \log_b(x) + 3\log_b(y) \).

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Most popular questions from this chapter

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log x+7 \log y $$

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