If log102 = 0.3010 and log103 = 0.4771, simplify: (Log108-log92) ÷ (Log109 - log102)

The question as printed has typos: "Log108 - log92" and "log109 - log102" mix bases 10 and 9. The intended expression is almost certainly:

(log₁₀8 − log₁₀2) ÷ (log₁₀9 − log₁₀2)

Let me check the options against that reading.

Step 1: Numerator

log 8 = 3 log 2 = 3(0.3010) = 0.9030

log 8 − log 2 = 0.9030 − 0.3010 = 0.6020

(Or: log(8/2) = log 4 = 2 log 2 = 0.6020.)

Step 2: Denominator

log 9 = 2 log 3 = 2(0.4771) = 0.9542

log 9 − log 2 = 0.9542 − 0.3010 = 0.6532

Step 3: Divide

0.6020 ÷ 0.6532 ≈ 0.9216

This matches none of the options (A to E). So under this reading, the question cannot give any listed answer.

Trying another reading: the "log" in the original is a subtraction typed as a division of different terms

Suppose the question is (log 8 − log 2) as a product or difference with the second bracket:

• 0.6020 − 0.6532 = −0.0512 (not an option)
• Try (log 8 − log 2) in the denominator form: 0.6532 ÷ 0.6020 ≈ 1.085 (not an option)

Try the reading log₁₀8 − log₁₀2 with log₁₀9 − log₁₀2 where the numerator uses 8 and 3 differently:

• log 8 = 0.9030 and log 9 = 0.9542; 0.9030 − 0.9542 = −0.0512 (not an option)

Another reading: (log 8 − log 9) ... using 2 and 3: 3(0.3010) − 2(0.4771) = 0.9030 − 0.9542 = −0.0512, still not matching −0.0784.

Check option E (−0.0784): this would need something like 0.3010×? … I could not reproduce −0.0784 from any sensible reading of the given data, so I cannot verify it.

Conclusion

0784). With the most sensible interpretation, the value is about 0.92, which isn't among the options. The question as typed seems to be corrupted, probably from the base notation (log₉ and log₂ with the subscripts misplaced). Please check your original paper or textbook for the exact wording. If you can share it, I'll work through it again.

Tip: In log questions, always break numbers into prime factors (8 = 2³, 9 = 3²) so you can use the given values of log 2 and log 3.

Since the official key says E and I could not derive it, E is the only option I can offer, but I can't confirm it from my own working.