is it true that the x-intercept of y=loga(x) is (1,0) for all positive constants a (except a=1)?

Yes, it is true that the x-intercept of the function \( y = \log_a(x) \) is \( (1,0) \) for all positive constants \( a \) (except \( a = 1 \)).

Here's why:

1. **Definition of the logarithmic function:**  
   The function \( y = \log_a(x) \) represents the logarithm of \( x \) with base \( a \), and it is the inverse of the exponential function \( y = a^x \). This means \( \log_a(x) \) gives the power to which the base \( a \) must be raised to get \( x \).
   
2. **Finding the x-intercept:**  
   The x-intercept occurs when \( y = 0 \). For the function \( y = \log_a(x) \), this means:
   \[
   \log_a(x) = 0
   \]
   By the definition of logarithms, this equation can be rewritten as:
   \[
   a^0 = x
   \]
   Since \( a^0 = 1 \) for any positive constant \( a \), we have:
   \[
   x = 1
   \]
   Therefore, the x-intercept is \( (1, 0) \), regardless of the base \( a \), as long as \( a > 0 \) and \( a \neq 1 \).

3. **Why \( a \neq 1 \):**  
   If \( a = 1 \), the logarithmic function becomes undefined because \( \log_1(x) \) is not valid for any \( x > 0 \). The base of a logarithm must be a positive number different from 1.

### Summary:
For any base \( a > 0 \) and \( a \neq 1 \), the x-intercept of the function \( y = \log_a(x) \) is always \( (1, 0) \).

Do you want more details or have any questions?

Here are 5 related questions:

1. How do the shapes of the graphs of \( y = \log_a(x) \) change with different values of \( a \)?
2. What is the domain and range of \( y = \log_a(x) \) for \( a > 1 \)?
3. What happens to the graph of \( y = \log_a(x) \) when \( 0 < a < 1 \)?
4. Can you explain the behavior of \( y = \log_a(x) \) as \( x \to 0^+ \) and as \( x \to \infty \)?
5. How does the x-intercept of \( y = \log_a(x) \) compare with the y-intercept of the inverse function \( y = a^x \)?

**Tip:** Always remember that the base of a logarithm must be positive and not equal to 1 for the function to be valid.

Is it true that the x-intercept of y=loga(x) is (1,0) for all positive constants a (except a=1)?

Discover why the x-intercept of the function y = log_a(x) is always (1,0) for any positive constant a, except when a = 1. This problem explores key logarithmic concepts, including inverse functions and exponential relationships, ideal for students in Grades 9-12.

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