Evaluate log base 9 of 0.4

Problem

(log_9)(0.4)

Solution

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

(log_9)(0.4)=ln(0.4)/ln(9)

2. Convert the decimal to a fraction to simplify the expression further.

0.4=4/10=2/5

3. Substitute the fraction back into the change of base formula.

(log_9)(0.4)=ln(2/5)/ln(9)

4. Apply the quotient rule for logarithms to the numerator.

ln(2/5)=ln(2)−ln(5)

5. Rewrite the denominator by expressing 9 as a power of 3.

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

6. Combine the results to find the exact value.

(log_9)(0.4)=(ln(2)−ln(5))/(2*ln(3))

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

ln(0.4)≈−0.91629

ln(9)≈2.19722

(−0.91629)/2.19722≈−0.4170

Final Answer

(log_9)(0.4)≈−0.4170

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