Problem 12
Question
Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{5}\left(\frac{125}{y}\right)$$
Step-by-Step Solution
Verified Answer
The expanded form of \( \log _{5}(\frac{125}{y}) \) is \( 3 - \log_5 y \).
1Step 1: Recognize the Components
Identify and separate the two components within the logarithm: 125 and y. The base of the logarithm is 5.
2Step 2: Apply the Quotient Rule
Apply the rule of logarithms that states \( log_b (\frac{m}{n}) = log_b m - log_b n \). Thus, \( log_5 (\frac{125}{y}) \) can be expanded as \( log_5 125 - log_5 y \)
3Step 3: Address the Logarithm
To further simplify the logarithm, rewrite 125 as \(5^3\). Because the base of the logarithm is 5, \( log_5 5^3 \) equals 3 according to the identity \( log_b b^x = x \). Thus, \( log_5 125 \) becomes 3.
4Step 4: Final Result
Combine the results of the two previous steps to write the fully expanded form of the original problem. The expanded expression is \( 3 - log_5 y \).
Other exercises in this chapter
Problem 11
Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calcula
View solution Problem 11
Write each equation in its equivalent logarithmic form. $$2^{-4}=\frac{1}{16}$$
View solution Problem 12
Write each equation in its equivalent logarithmic form. $$5^{-3}=\frac{1}{125}$$
View solution Problem 13
Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$3^{1-x}=\frac{1}{27}$$
View solution