Evaluate log base 2 of 12

Problem

(log_2)(12)

Solution

1. Apply the change of base formula to rewrite the expression using common logarithms (base 10) or natural logarithms (base e.

(log_b)(a)=(log_c)(a)/(log_c)(b)

2. Substitute the values a=12 and b=2 into the formula using natural logarithms.

(log_2)(12)=ln(12)/ln(2)

3. Factor the number 12 into 2⋅3 to simplify the expression using logarithm properties.

(log_2)(12)=(log_2)(2⋅3)

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

(log_2)(12)=(log_2)(2)+(log_2)(3)

5. Simplify the terms using the power rule (log_b)(xk)=k*(log_b)(x) and the identity (log_b)(b)=1

(log_2)(12)=2*(log_2)(2)+(log_2)(3)

(log_2)(12)=2*(1)+(log_2)(3)

(log_2)(12)=2+(log_2)(3)

6. Calculate the numerical value using the change of base result from step 2.

ln(12)/ln(2)≈2.4849/0.6931

(log_2)(12)≈3.585

Final Answer

(log_2)(12)=2+(log_2)(3)≈3.585

Want more problems? Check here!