Evaluate log base 4 of 20

Problem

(log_4)(20)

Solution

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

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

2. Substitute the given values into the formula.

(log_4)(20)=ln(20)/ln(4)

3. Rewrite the argument of the numerator to simplify the expression further using properties of logarithms.

(log_4)(20)=ln(4⋅5)/ln(4)

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

(log_4)(20)=(ln(4)+ln(5))/ln(4)

5. Split the fraction to simplify the terms.

(log_4)(20)=ln(4)/ln(4)+ln(5)/ln(4)

6. Simplify the first term and express the second term back in base 4 notation if needed, or calculate the decimal approximation.

(log_4)(20)=1+(log_4)(5)

7. Calculate the numerical value using a calculator for the final decimal result.

(log_4)(20)≈2.160964

Final Answer

(log_4)(20)=ln(20)/ln(4)≈2.1610

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