Evaluate log base 3 of 12

Problem

(log_3)(12)

Solution

1. Apply the change of base formula to rewrite the expression using natural logarithms or common logarithms.

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

2. Substitute the values a=12 and b=3 into the formula.

(log_3)(12)=ln(12)/ln(3)

3. Factor the number 12 into 3×4 to simplify the expression further using logarithm properties.

(log_3)(12)=(log_3)(3×4)

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

(log_3)(12)=(log_3)(3)+(log_3)(4)

5. Simplify the term (log_3)(3) since any log with the same base and argument equals 1.

(log_3)(12)=1+(log_3)(4)

6. Rewrite the final expression using the change of base formula for a numerical approximation if needed.

(log_3)(12)=1+ln(4)/ln(3)

Final Answer

(log_3)(12)=1+(log_3)(4)

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