Coefficient of `x^10` in `10^x` is

Options

A. `((log_e 10)^(10))/(10!)`
B. `1/(10!)`
C. `((log_(10) e)^(10))/(10!)`
D. `1/10`

Answer

To find the coefficient of \( x^{10} \) in the expansion of \( 10^x \), we can follow these steps:

### Step 1: Rewrite \( 10^x \) using the exponential function
We start by expressing \( 10^x \) in terms of the natural exponential function:
\[ 10^x = e^{x \ln(10)} \]
Here, \( \ln(10) \) is the natural logarithm of 10.

### Step 2: Use the Taylor series expansion for \( e^x \)
The Taylor series expansion for \( e^x \) is given by:
\[ e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \]
Substituting \( x \ln(10) \) into this series, we have:
\[ e^{x \ln(10)} = 1 + \frac{x \ln(10)}{1!} + \frac{(x \ln(10))^2}{2!} + \frac{(x \ln(10))^3}{3!} + \cdots \]

### Step 3: Identify the general term in the series
The general term in the expansion can be expressed as:
\[ \frac{(x \ln(10))^n}{n!} \]
where \( n \) is the term number.

### Step 4: Find the coefficient of \( x^{10} \)
To find the coefficient of \( x^{10} \), we need to look for the term where \( n = 10 \):
\[ \frac{(x \ln(10))^{10}}{10!} = \frac{x^{10} (\ln(10))^{10}}{10!} \]
Thus, the coefficient of \( x^{10} \) is:
\[ \frac{(\ln(10))^{10}}{10!} \]

### Final Answer
The coefficient of \( x^{10} \) in the expansion of \( 10^x \) is:
\[ \frac{(\ln(10))^{10}}{10!} \]

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