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Chapter 3: Problem 53

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$2 \log _{b} x+3 \log _{b} y$$

Short Answer

Expert verified
The single logarithm equivalent to the given expression is \(\log_b (x^2y^3)\)

Step by step solution

01

Identify the terms that can be combined using Logarithm's properties

In this case, the given expression is \(2 \log _{b} x+3 \log _{b} y\). The base for both logarithms is 'b', so they can certainly be combined.
02

Apply the Power Rule of Logarithms

According to the power rule, \(a \log_b X = \log_b X^a\). Applying this rule, we have: \(2 \log _{b} x+3 \log _{b} y = \log _{b} x^2+\log _{b} y^3\)
03

Use the Product Rule of Logarithms

According to the product rule, \(\log_b X + \log_b Y = \log_b (XY)\). Using this rule, the expression becomes \(\log_b (x^2y^3)\)

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