To find the intersection point between the function [tex]\( f(x) = \log_3(x) \)[/tex] and the line [tex]\( y = 1 \)[/tex], we need to determine the x-coordinate where the function [tex]\( f(x) \)[/tex] equals 1.
The function [tex]\( f(x) = \log_3(x) \)[/tex] represents the logarithm of [tex]\( x \)[/tex] with base 3. We are looking for the value of [tex]\( x \)[/tex] that makes this equation true:
[tex]\[ \log_3(x) = 1 \][/tex]
By definition of logarithms, [tex]\( \log_b(a) = c \)[/tex] means [tex]\( b^c = a \)[/tex]. Thus, we can rewrite the equation [tex]\( \log_3(x) = 1 \)[/tex] in exponential form:
[tex]\[ 3^1 = x \][/tex]
Solving this, we find:
[tex]\[ x = 3 \][/tex]
Therefore, the x-coordinate of the intersection point is [tex]\( x = 3 \)[/tex]. Since we are given that [tex]\( y = 1 \)[/tex] from the line equation [tex]\( y = 1 \)[/tex], the intersection point in the coordinate plane is:
[tex]\[ (3, 1) \][/tex]
So, the correct answer is [tex]\((3, 1)\)[/tex].
