Given that a = log 7 and b = $\log$ 2, express log 35 in terms of a and b.

I'll work this out directly. The log base isn't stated, so I'm assuming base 10, which is the standard for WASSCE.

Analyze the question We need log 35 written using a = log 7 and b = log 2.

Key concepts

• Product rule: log(xy) = log x + log y
• Quotient rule: log(x/y) = log x − log y
• In base 10, log 10 = 1

Solution

Factorise 35: 35 = 7 × 5

So log 35 = log 7 + log 5 = a + log 5

Now express 5 using 2 and 10: 5 = 10 / 2

log 5 = log 10 − log 2 = 1 − b

Therefore: log 35 = a + (1 − b) = a − b + 1

Check with numbers: log 7 ≈ 0.8451 and log 2 ≈ 0.3010 a − b + 1 = 0.8451 − 0.3010 + 1 = 1.5441 log 35 ≈ 1.5441 ✓

Comparing the options

• A (a + b + 1) gives about 2.146. Wrong.
• B (ab − 1) gives about −0.746. Wrong.
• C (a − b + 1) gives about 1.544. Correct.
• D (b − a + 1) gives about 0.456. Wrong.

Exam tip: When you see log 5 and are given log 2, remember that 5 = 10/2, so log 5 = 1 − log 2. This trick appears often in JAMB and WASSCE. Similarly, log 25 = 2 − 2 log 2, and log 20 = 1 + log 2.