Evaluate log base 4 of 125

Problem

(log_4)(125)

Solution

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

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

2. Substitute the values into the formula where a=125 and b=4

(log_4)(125)=ln(125)/ln(4)

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

125=5

4=2

4. Substitute the powers back into the logarithmic expression.

(log_4)(125)=ln(5)/ln(2)

5. Apply the power rule for logarithms, ln(xk)=k*ln(x) to move the exponents to the front.

(log_4)(125)=(3*ln(5))/(2*ln(2))

6. Calculate the decimal approximation by evaluating the logarithms.

ln(5)≈1.6094379

ln(2)≈0.6931472

(log_4)(125)≈(3⋅1.6094379)/(2⋅0.6931472)

(log_4)(125)≈3.482892

Final Answer

(log_4)(125)=(3*ln(5))/(2*ln(2))≈3.482892

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