Evaluate log base 4 of 1/256

Problem

(log_4)(1/256)

Solution

1. Identify the base and the argument of the logarithm. The base is 4 and the argument is 1/256

2. Rewrite the argument as a power of the base. Since 4=256 it follows that 256 can be expressed as 4

3. Apply the rule for negative exponents, which states that 1/(an)=a(−n)

1/256=1/4

1/256=4(−4)

4. Substitute the power back into the logarithmic expression.

(log_4)(4(−4))

5. Use the logarithm property (log_b)(bx)=x to evaluate the expression.

(log_4)(4(−4))=−4

Final Answer

(log_4)(1/256)=−4

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