The greatest possible natural number n that satisfies the inequality 9^n < 10^8, given that log 3 = 0.4771 and n ∈ N, is equal to:

Introduction / Context:
This question involves comparing an exponential expression with a power of 10 and finding the largest natural number n that satisfies a given inequality. Logarithms are provided as a tool for handling powers, particularly because the base of the exponent is 9 and the comparison base is 10. Problems of this form are common in aptitude tests to check fluency with logarithmic transformations and inequalities.

Given Data / Assumptions:
We need the largest natural number n such that 9^n < 10^8.We are given log 3 = 0.4771 (base 10).The variable n is a natural number, so n is a positive integer.We use base 10 logarithms for convenience.

Concept / Approach:
The inequality compares 9^n with 10^8. Taking logarithms on both sides turns powers into products, which makes it easier to isolate n. Since 9 = 3^2, we can rewrite 9^n as 3^(2n). Using log rules, log (a^k) = k log a. The inequality then becomes linear in n, and we can solve for n using the given logarithm of 3. Finally, as n must be an integer, we choose the largest integer that still satisfies the inequality.

Step-by-Step Solution:
Step 1: Rewrite 9^n using base 3: 9 = 3^2, so 9^n = (3^2)^n = 3^(2n).Step 2: The inequality becomes 3^(2n) < 10^8.Step 3: Take base 10 logarithms of both sides: log(3^(2n)) < log(10^8).Step 4: Apply log power rule: 2n log 3 < 8 log 10.Step 5: Since log 10 = 1 in base 10, this simplifies to 2n log 3 < 8.Step 6: Substitute the given value log 3 = 0.4771: 2n * 0.4771 < 8.Step 7: Compute 2 * 0.4771 = 0.9542, so 0.9542 n < 8.Step 8: Solve for n: n < 8 / 0.9542, which is approximately n < 8.38.Step 9: Since n must be a natural number, the greatest integer less than 8.38 is 8.

Verification / Alternative check:
As a check, evaluate 9^8 and 9^9. The value 9^8 is 43046721, which is less than 10^8 (that is, 100000000). The value 9^9 is 387420489, which is greater than 10^8. Therefore 9^8 < 10^8 but 9^9 ≥ 10^8. This confirms that n = 8 is the greatest natural number that satisfies the inequality.

Why Other Options Are Wrong:
The option 7 is too small; it satisfies the inequality but is not the greatest possible value. The options 9 and 10 suggest values that produce powers of 9 that are larger than 10^8, so they violate the inequality. Only n = 8 satisfies the inequality and is the largest such natural number.

Common Pitfalls:
One frequent mistake is to ignore the requirement that n be a natural number and to give a decimal answer like 8.38. Another error is to forget that when taking logarithms, the direction of the inequality remains unchanged because log is an increasing function for positive numbers. Some candidates also forget that log 10 is equal to 1 in base 10, which can lead to incorrect simplification.

Final Answer:
The greatest possible natural number n that satisfies 9^n < 10^8 is 8.