Number of positive integral value(s) of `a` for which the curve `y=a^(x)` intersects the line `y=x` is:

Options

A. (a) `0`
B. (b) `1`
C. (c) `2`
D. (d) more than `2`

Answer

To find the number of positive integral values of \( a \) for which the curve \( y = a^x \) intersects the line \( y = x \), we can follow these steps:

### Step 1: Set up the equation for intersection
To find the points of intersection between the curve and the line, we set the equations equal to each other:
\[ a^x = x \]

### Step 2: Take the logarithm of both sides
Taking the logarithm of both sides helps us to manipulate the equation:
\[ \log(a^x) = \log(x) \]
Using the property of logarithms, we can rewrite the left side:
\[ x \log(a) = \log(x) \]

### Step 3: Rearrange the equation
Now we can rearrange the equation:
\[ x \log(a) - \log(x) = 0 \]

### Step 4: Analyze the function
Define a function based on the rearranged equation:
\[ f(x) = x \log(a) - \log(x) \]
We want to find the values of \( a \) such that this function has at least one positive solution \( x \).

### Step 5: Determine the behavior of the function
To analyze the function \( f(x) \):
- As \( x \to 0^+ \), \( \log(x) \to -\infty \) and thus \( f(x) \to +\infty \).
- As \( x \to \infty \), \( \log(x) \) grows slower than \( x \log(a) \) if \( \log(a) > 0 \).

### Step 6: Find critical points
To find critical points, we differentiate \( f(x) \):
\[ f'(x) = \log(a) - \frac{1}{x} \]
Setting \( f'(x) = 0 \) gives:
\[ \log(a) = \frac{1}{x} \implies x = \frac{1}{\log(a)} \]

### Step 7: Determine the number of intersections
Now we need to check the conditions under which \( f(x) = 0 \) has solutions:
- If \( a = 1 \), then \( f(x) = 0 \) for \( x = 1 \).
- If \( a > 1 \), the function \( f(x) \) will have exactly one intersection with the x-axis.
- If \( 0 < a < 1 \), the function \( f(x) \) will not intersect the x-axis since \( a^x \) will always be less than \( x \) for positive \( x \).

### Step 8: Conclusion
The only positive integral value of \( a \) that allows \( y = a^x \) to intersect \( y = x \) is \( a = 1 \).

Thus, the number of positive integral values of \( a \) for which the curve intersects the line is:
\[ \boxed{1} \]