Simplify/Condense 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 rewrite them in terms of natural logarithms.

(log_2)(6)=ln(6)/ln(2)

(log_6)(8)=ln(8)/ln(6)

2. Multiply the expressions together and observe that the common term ln(6) appears in both the numerator and the denominator.

ln(6)/ln(2)⋅ln(8)/ln(6)

3. Cancel the common factor ln(6) from the expression.

ln(8)/ln(2)

4. Rewrite the numerator by expressing 8 as a power of 2, specifically 8=2

ln(2)/ln(2)

5. Apply the power rule for logarithms, which states ln(ab)=b*ln(a) to move the exponent to the front.

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

6. Simplify the fraction by canceling the common term ln(2)

3

Final Answer

(log_2)(6)⋅(log_6)(8)=3

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