Evaluate log base 8 of 1/4

Problem

(log_8)(1/4)

Solution

1. Set the expression equal to a variable x to represent the unknown value.

(log_8)(1/4)=x

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

8^x=1/4

3. Express both sides with a common base of 2 noting that 8=2^3 and 4=2^2

(2^3)^x=1/2^2

4. Apply exponent rules to simplify both sides, specifically (a^m)^n=a^(m*n) and 1/(a^n)=a^(−n)

2^(3*x)=2^(−2)

5. Equate the exponents since the bases are now identical.

3*x=−2

6. Solve for x by dividing both sides by 3

x=−2/3

Final Answer

(log_8)(1/4)=−2/3

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