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Chapter 3: Problem 64

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $$\log _{10} \frac{x y^{4}}{z^{5}}$$

Short Answer

Expert verified
The expanded form of the given logarithmic expression is \( \log _{10} x + 4 \log _{10} y - 5 \log _{10} z \)

Step by step solution

01

Apply the quotient rule of logarithm

The quotient rule of logarithm states that the logarithm of a quotient is the difference of the logarithms of the numerator and denominator. So the given expression can be rewritten as: \( \log _{10} (x y^{4}) - \log _{10} (z^{5}) \)
02

Apply the product rule of logarithm

The product rule states that the logarithm of a product is the sum of the logarithms of the factors. Apply this rule to expand \( \log _{10} (x y^{4}) \) into \( \log _{10} x + \log _{10} y^{4} \). The expression from Step 1 thus becomes: \( \log _{10} x + \log _{10} y^{4} - \log _{10} (z^{5}) \)
03

Apply the power rule of logarithm

The power rule states that the logarithm of a number to a certain power is the product of the power and the logarithm of the number itself. It is applied to \( \log _{10} y^{4} \) and \( \log _{10} z^{5} \) to get \( 4 \log _{10} y \) and \( 5 \log _{10} z \) respectively. So, the final expanded form of the given logarithmic expression is: \( \log _{10} x + 4 \log _{10} y - 5 \log _{10} z \)

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Most popular questions from this chapter

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