Evaluate log base 2 of 9

Problem

(log_2)(9)

Solution

1. Apply the change of base formula to rewrite the logarithm in terms of common logarithms (base 10) or natural logarithms (base e.

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

2. Substitute the values a=9 and b=2 into the formula.

(log_2)(9)=ln(9)/ln(2)

3. Rewrite the number 9 as 3 to simplify the expression using logarithm properties.

(log_2)(9)=ln(3)/ln(2)

4. Apply the power rule for logarithms, which states ln(xy)=y*ln(x)

(log_2)(9)=(2*ln(3))/ln(2)

5. Calculate the numerical value using the approximate values ln(3)≈1.0986 and ln(2)≈0.6931

(log_2)(9)≈(2*(1.0986))/0.6931

6. Simplify the quotient to find the final decimal approximation.

(log_2)(9)≈3.1699

Final Answer

(log_2)(9)=ln(9)/ln(2)≈3.1699

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