Evaluate log base 5 of 10

Problem

(log_5)(10)

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 given values into the formula.

(log_5)(10)=ln(10)/ln(5)

3. Simplify the numerator using the property of logarithms ln(a*b)=ln(a)+ln(b) to see the relationship between the terms.

ln(10)=ln(2⋅5)=ln(2)+ln(5)

4. Divide the terms to find the exact simplified form.

(ln(2)+ln(5))/ln(5)=ln(2)/ln(5)+1

5. Calculate the approximate decimal value using a calculator if required.

ln(10)/ln(5)≈1.430676558

Final Answer

(log_5)(10)=ln(10)/ln(5)

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