Evaluate log base 3 of 0.015

Problem

(log_3)(0.015)

Solution

1. Convert the decimal to a fraction to make the expression easier to manipulate.

(log_3)(0.015)=(log_3)(15/1000)

2. Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 5.

15/1000=3/200

3. Apply the quotient rule for logarithms, which states that (log_b)(x/y)=(log_b)(x)−(log_b)(y)

(log_3)(3/200)=(log_3)(3)−(log_3)(200)

4. Evaluate the first term using the property (log_b)(b)=1

(log_3)(3)=1

5. Apply the change of base formula, (log_b)(a)=ln(a)/ln(b) to find the numerical value of the expression.

(log_3)(0.015)=ln(0.015)/ln(3)

6. Calculate the final decimal value using a calculator.

(−4.199705...)/(1.098612...)≈−3.8227

Final Answer

(log_3)(0.015)≈−3.8227

Want more problems? Check here!