Evaluate log base 0.5 of 64

Problem

(log_0.5)(64)

Solution

1. Identify the logarithmic equation by setting the expression equal to x

(log_0.5)(64)=x

2. Rewrite the equation in exponential form using the definition (log_b)(y)=x⇔bx=y

0.5=64

3. Express the base 0.5 and the number 64 as powers of 2

0.5=1/2=2(−1)

64=2

4. Substitute these powers back into the exponential equation.

(2(−1))x=2

5. Apply the power of a power rule (am)n=a(m*n) to simplify the left side.

2(−x)=2

6. Equate the exponents since the bases are the same.

−x=64

−x=6

7. Solve for x by multiplying both sides by −1

x=−6

Final Answer

(log_0.5)(64)=−6

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