The number of solution(s) of the equation `sinx = log_(10) x` is/are

Explanation

To find the number of solutions for the equation \( \sin x = \log_{10} x \), we will analyze both sides of the equation graphically and mathematically.

### Step 1: Understand the functions involved
The left side of the equation is \( \sin x \), which oscillates between -1 and 1 for all \( x \). The right side, \( \log_{10} x \), is defined for \( x > 0 \) and increases from \( -\infty \) to \( +\infty \) as \( x \) increases from 0 to \( +\infty \).

**Hint:** Remember that \( \sin x \) is periodic, while \( \log_{10} x \) is a monotonically increasing function.

### Step 2: Analyze the range of \( \log_{10} x \)
For \( x = 1 \), \( \log_{10} 1 = 0 \). As \( x \) approaches 0 from the right, \( \log_{10} x \) approaches \( -\infty \). As \( x \) increases beyond 1, \( \log_{10} x \) becomes positive and continues to increase.

**Hint:** Note the behavior of \( \log_{10} x \) as \( x \) approaches 0 and as \( x \) increases.

### Step 3: Determine the intersections
Since \( \sin x \) oscillates between -1 and 1, we need to find where \( \log_{10} x \) intersects with \( \sin x \) within this range.

- For \( x < 1 \), \( \log_{10} x < 0 \) and \( \sin x \) can be negative or positive.
- For \( x = 1 \), \( \log_{10} 1 = 0 \) and \( \sin 1 \) is positive (approximately 0.84).
- For \( x > 1 \), \( \log_{10} x \) increases and will eventually surpass 1, but \( \sin x \) will oscillate.

**Hint:** Look for the points where \( \log_{10} x \) crosses the x-axis and where it stays within the bounds of \( \sin x \).

### Step 4: Graphical representation
By plotting the graphs of \( \sin x \) and \( \log_{10} x \):
- The graph of \( \sin x \) oscillates between -1 and 1.
- The graph of \( \log_{10} x \) starts from \( -\infty \) when \( x \) is close to 0, crosses the x-axis at \( x = 1 \), and continues to rise.

From the graphical analysis, we can observe that there is only one point where \( \sin x \) intersects \( \log_{10} x \) in the interval \( (0, +\infty) \).

### Step 5: Conclusion
Thus, the number of solutions to the equation \( \sin x = \log_{10} x \) is **1**.

**Final Answer:** The number of solutions is **1**.