Evaluate log base 8 of 3

Problem

(log_8)(3)

Solution

1. Apply the change of base formula to rewrite the logarithm using a more common base, such as the natural logarithm ln() or common logarithm (log_10)()

(log_b)(a)=ln(a)/ln(b)

2. Substitute the values from the expression into the formula, where a=3 and b=8

(log_8)(3)=ln(3)/ln(8)

3. Simplify the denominator by recognizing that 8=2 and applying the power rule for logarithms ln(xy)=y*ln(x)

ln(8)=ln(2)=3*ln(2)

4. Combine the results to express the final exact value.

(log_8)(3)=ln(3)/(3*ln(2))

Final Answer

(log_8)(3)=ln(3)/ln(8)

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