Questions: What is the value of log base 5 of 1/125?

Transcript text: What is the value of $\log _{5} \frac{1}{125}$ ?

Solution

Solution Steps

Step 1: Identify the Logarithmic Expression

We need to evaluate the expression $ \log_{5} \frac{1}{125} $.

Step 2: Rewrite the Argument

Recognize that $ 125 $ can be expressed as a power of $ 5 $: \[ 125 = 5^3 \] Thus, we can rewrite $ \frac{1}{125} $ as: \[ \frac{1}{125} = 5^{-3} \]

Step 3: Apply the Logarithmic Identity

Using the property of logarithms, we have: \[ \log_{b} (a^c) = c \cdot \log_{b} (a) \] Applying this to our expression: \[ \log_{5} \left( 5^{-3} \right) = -3 \cdot \log_{5} (5) \] Since $ \log_{5} (5) = 1 $, we find: \[ \log_{5} \left( 5^{-3} \right) = -3 \cdot 1 = -3 \]

Final Answer

Thus, the value of $ \log_{5} \frac{1}{125} $ is \[ \boxed{-3} \]

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