Find the 10th term in the expansion of `5^x`

Answer

To find the 10th term in the expansion of \(5^x\), we can use the concept of the Taylor series expansion. Here’s a step-by-step solution:

### Step-by-Step Solution:

1. **Understand the Expression**:
We start with the expression \(5^x\). We can rewrite this using the exponential function:
\[ 5^x = e^{x \log 5} \]

2. **Use the Taylor Series Expansion**:
The Taylor series expansion for \(e^y\) around \(y=0\) is given by:
\[ e^y = \sum_{n=0}^{\infty} \frac{y^n}{n!} \]
In our case, \(y = x \log 5\). Therefore, we can write:
\[ e^{x \log 5} = \sum_{n=0}^{\infty} \frac{(x \log 5)^n}{n!} \]

3. **Identify the 10th Term**:
The \(n\)-th term in the expansion is given by:
\[ T_n = \frac{(x \log 5)^n}{n!} \]
For the 10th term, we need to find \(T_9\) (since we start counting from \(n=0\)):
\[ T_9 = \frac{(x \log 5)^9}{9!} \]

4. **Final Expression**:
Thus, the 10th term in the expansion of \(5^x\) is:
\[ T_{10} = \frac{(x \log 5)^9}{9!} \]