Evaluate log base 3 of 6^3*9^6

Problem

(log_3)(6⋅9)

Solution

1. Apply the product rule for logarithms, which states that (log_b)(x*y)=(log_b)(x)+(log_b)(y)

(log_3)(6⋅9)=(log_3)(6)+(log_3)(9)

2. Apply the power rule for logarithms, which states that (log_b)(xp)=p*(log_b)(x)

(log_3)(6)+(log_3)(9)=3*(log_3)(6)+6*(log_3)(9)

3. Decompose the arguments into prime factors base 3 where possible. Note that 6=2⋅3 and 9=3

3*(log_3)(2⋅3)+6*(log_3)(3)

4. Expand the first term using the product rule and simplify the second term using the property (log_b)(bx)=x

3*((log_3)(2)+(log_3)(3))+6*(2)

5. Simplify the constants knowing that (log_3)(3)=1

3*((log_3)(2)+1)+12

6. Distribute and combine the numerical terms to reach the final expression.

3*(log_3)(2)+3+12

3*(log_3)(2)+15

Final Answer

(log_3)(6⋅9)=3*(log_3)(2)+15

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