Questions: Find the logarithm. log4(1/64)=

Transcript text: Find the logarithm. \[ \log _{4}\left(\frac{1}{64}\right)= \]

Solution

Solution Steps

Step 1: Understanding the Logarithm

We need to solve the logarithmic equation: \[ \log_{4}\left(\frac{1}{64}\right) = x \] This means we are looking for \( x \) such that: \[ 4^x = \frac{1}{64} \]

Step 2: Expressing the Fraction

We can rewrite \( \frac{1}{64} \) as: \[ \frac{1}{64} = 64^{-1} \] Since \( 64 = 4^3 \), we have: \[ \frac{1}{64} = (4^3)^{-1} = 4^{-3} \]

Step 3: Equating the Exponents

From the equation \( 4^x = 4^{-3} \), we can equate the exponents: \[ x = -3 \]

Final Answer

Thus, the solution to the logarithmic equation is: \[ \boxed{x = -3} \]

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