Evaluate log base 2 of 6* log base 6 of 8

Problem

(log_2)(6)⋅(log_6)(8)

Solution

1. Apply the change of base formula to both logarithmic terms to express them in terms of natural logarithms.

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

2. Rewrite the expression using the change of base formula.

ln(6)/ln(2)⋅ln(8)/ln(6)

3. Cancel the common factor of ln(6) from the numerator and the denominator.

ln(8)/ln(2)

4. Express the number 8 as a power of 2.

8=2

5. Substitute the power back into the expression.

ln(2)/ln(2)

6. Apply the power rule for logarithms, which states ln(xk)=k*ln(x)

(3*ln(2))/ln(2)

7. Simplify the fraction by canceling ln(2)

3

Final Answer

(log_2)(6)⋅(log_6)(8)=3

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