Evaluate 3 log 5 + 2 log 4 − log 2, where log denotes the common logarithm…
2024
Evaluate 3 log 5 + 2 log 4 − log 2, where log denotes the common logarithm (base 10).
Answer: C. 3 — ConceptFor logarithms of the same base, a coefficient becomes an exponent: k log a = log(ak). The product and quotient laws are log a + log b = log(ab) and…
- A.
6
- B.
9
- C.
3
- D.
5
Attempted by 11 students.
Show answer & explanation
Correct answer: C
Concept
For logarithms of the same base, a coefficient becomes an exponent: k log a = log(ak).
The product and quotient laws are log a + log b = log(ab) and log a − log b = log(a/b).
Application
Use the power law: 3 log 5 = log(53) and 2 log 4 = log(42).
Combine the terms: log(53) + log(42) − log 2 = log[(125 × 16)/2].
Simplify the argument: (125 × 16)/2 = 125 × 8 = 1000.
Therefore the expression is log 1000 = log(103) = 3.
Cross-check
Expanding directly, 3 log 5 + 2 log 4 − log 2 = log 125 + log 16 − log 2 = log 1000, confirming the value 3.
Thus, the value of the expression is 3.