The greatest possible value of n could be if `9^(n)lt10^(8)`, given that `log3=0.4771` and `n in N` :

Options

A. A) 7
B. B) 8
C. C) 9
D. D) 10

Answer

To solve the inequality \( 9^n < 10^8 \) where \( n \) is a natural number, we can follow these steps:

### Step 1: Take the logarithm of both sides
We start with the inequality:
\[ 9^n < 10^8 \]
Taking the logarithm (base 10) of both sides gives:
\[ \log(9^n) < \log(10^8) \]

### Step 2: Apply logarithmic properties
Using the property of logarithms that states \( \log(a^b) = b \cdot \log(a) \), we can rewrite the inequality:
\[ n \cdot \log(9) < 8 \cdot \log(10) \]

### Step 3: Simplify the right side
Since \( \log(10) = 1 \), we can simplify the right side:
\[ n \cdot \log(9) < 8 \]

### Step 4: Express \( \log(9) \) in terms of \( \log(3) \)
We know that \( 9 = 3^2 \), so we can express \( \log(9) \) as:
\[ \log(9) = \log(3^2) = 2 \cdot \log(3) \]
Substituting this into our inequality gives:
\[ n \cdot (2 \cdot \log(3)) < 8 \]

### Step 5: Isolate \( n \)
Now we can isolate \( n \):
\[ n < \frac{8}{2 \cdot \log(3)} \]
This simplifies to:
\[ n < \frac{8}{2 \cdot 0.4771} \]
Calculating the denominator:
\[ 2 \cdot 0.4771 = 0.9542 \]
So we have:
\[ n < \frac{8}{0.9542} \]

### Step 6: Calculate the right side
Now we calculate:
\[ \frac{8}{0.9542} \approx 8.38 \]

### Step 7: Determine the greatest natural number \( n \)
Since \( n \) must be a natural number, the greatest possible value of \( n \) is:
\[ n = 8 \]

### Final Answer
Thus, the greatest possible value of \( n \) is:
\[ \boxed{8} \]