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Chapter 12: Problem 21

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{b}\left(x^{2} y\right)$$

Short Answer

Expert verified
The expanded form of the expression \(\log_b(x^2y)\) is \(2 \cdot \log_b(x) + \log_b(y)\).

Step by step solution

01

Apply the Product Rule

The expression is the logarithm of a product, so start by applying the product rule for logarithms, which states that \(\log_b(xy) = \log_b(x) + \log_b(y)\). So, \(\log_b(x^2y) = \log_b(x^2) + \log_b(y)\)
02

Apply the Power Rule

The first term on the right side, \(\log_b(x^2)\), can further be simplified using the power rule of logarithms, \(\log_b(a^n) = n \cdot \log_b(a)\). So, \(\log_b(x^2) = 2 \cdot \log_b(x)\). Now, the log expression can be rewritten as \(2 \cdot \log_b(x) + \log_b(y)\)
03

Result

The expanded form of the given logarithmic expression, \(\log_b(x^2y)\), is \(2 \cdot \log_b(x) + \log_b(y)\).

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Most popular questions from this chapter

The function \(P(t)=145 e^{-0.092 t}\) models a runner's pulse, \(P(t),\) in beats per minute, \(t\) minutes after a race, where \(0 \leq t \leq 15 .\) Graph the function using a graphing utility. [TRACE] along the graph and determine after how many minutes the runner's pulse will be 70 beats per minute. Round to the nearest tenth of a minute. Verify your observation algebraically.

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