To determine the largest integer [tex]\( n \)[/tex] for which [tex]\( n^{200} < 5^{300} \)[/tex], let's follow a step-by-step approach using logarithmic properties.
### Step-by-Step Solution:
#### Step 1: Understanding the Inequality
We start with the inequality:
[tex]\[ n^{200} < 5^{300} \][/tex]
#### Step 2: Applying the Natural Logarithm
Taking natural logarithms on both sides of the inequality helps simplify the exponents:
[tex]\[ \ln(n^{200}) < \ln(5^{300}) \][/tex]
#### Step 3: Using Logarithmic Properties
Apply the property of logarithms that allows us to move the exponent in front of the logarithm:
[tex]\[ 200 \ln(n) < 300 \ln(5) \][/tex]
#### Step 4: Isolate the Logarithm of [tex]\( n \)[/tex]
Divide both sides by 200 to isolate [tex]\( \ln(n) \)[/tex]:
[tex]\[ \ln(n) < \frac{300 \ln(5)}{200} \][/tex]
#### Step 5: Calculate the Right-Hand Side
We need to evaluate the expression on the right-hand side:
[tex]\[ \frac{300 \ln(5)}{200} \approx 2.4141568686511503 \][/tex]
#### Step 6: Solving for [tex]\( n \)[/tex]
To find [tex]\( n \)[/tex], we use the fact that [tex]\( \ln(n) \approx 2.4141568686511503 \)[/tex]. Therefore, exponentiating both sides gives:
[tex]\[ n < e^{2.4141568686511503} \][/tex]
#### Step 7: Evaluate the Exponential
Calculate the value of [tex]\( e^{2.4141568686511503} \)[/tex]:
[tex]\[ e^{2.4141568686511503} \approx 11.18249396070347 \][/tex]
Since [tex]\( n \)[/tex] has to be an integer, the largest integer less than 11.1825 is 11.
### Conclusion
Thus, the largest integer [tex]\( n \)[/tex] for which [tex]\( n^{200} < 5^{300} \)[/tex] is:
[tex]\[ n = 11 \][/tex]
