If log 7 = 0.8451, log 3 = 0.4771, log 5 = 0.6990, log 2 = 0.3010, then find…

Concept

The power rule of logarithms states log an = n·log a, and the quotient rule states log(a/b) = log a − log b. Together these let a logarithm of a power of a fraction be rewritten as a multiple of the difference of two known logarithms.

Application

1. Convert the mixed number 2⅓ to the improper fraction 7/3.

2. Apply the power rule: log(7/3)5 = 5 log(7/3).

3. Apply the quotient rule: log(7/3) = log 7 − log 3.

4. Substitute the given values: 5(0.8451 − 0.4771).

5. Compute the result: 5 × 0.368 = 1.84.

Cross-check

Since 7/3 ≈ 2.333, and 100.368 ≈ 2.333, the intermediate value log(7/3) ≈ 0.368 is consistent, confirming 5 × 0.368 = 1.84 by a direct estimate of the fraction's logarithm.