Evaluate log base 4 of 9

Problem

(log_4)(9)

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=9 and b=4 into the formula.

(log_4)(9)=ln(9)/ln(4)

3. Rewrite the arguments as powers of prime numbers to simplify the expression.

(log_4)(9)=ln(3)/ln(2)

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

(log_4)(9)=(2*ln(3))/(2*ln(2))

5. Simplify the fraction by canceling the common factor of 2.

(log_4)(9)=ln(3)/ln(2)

6. Recognize the result as another change of base form or calculate the numerical value.

(log_4)(9)=(log_2)(3)

Final Answer

(log_4)(9)=ln(3)/ln(2)

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