Which of the following explains why $f(x)=\log _4 x$ does not have a $y$-intercept? Check all that apply.
$\checkmark$ There is no power of 4 that is equal to 0.
x There is no power of 4 that is equal to 1.
$\checkmark$ Its inverse does not have any $x$-intercepts.
x Its inverse does not have any $y$-intercepts.
The $x$-intercept of $f(x)=\log _4 x$ is $\square$
Answer (1)
f ( x ) = lo g 4 x does not have a y -intercept because there is no power of 4 that equals 0.
The inverse of f ( x ) , 4 x , does not have an x -intercept, which also explains why f ( x ) does not have a y -intercept.
To find the x -intercept, set f ( x ) = 0 and solve for x .
The x -intercept of f ( x ) = lo g 4 x is 1 .
Explanation
Understanding the Problem The function given is f ( x ) = lo g 4 x . We need to determine why this function does not have a y -intercept and then find its x -intercept.
Finding the Y-intercept The y -intercept of a function occurs where the graph of the function intersects the y -axis. This happens when x = 0 . So, we need to evaluate f ( 0 ) = lo g 4 0 . In other words, we are asking: 'To what power must we raise 4 to get 0?' Since any positive number raised to any power will never be 0, there is no power of 4 that equals 0. Therefore, f ( x ) = lo g 4 x does not have a y -intercept.
Considering the Inverse The inverse of f ( x ) = lo g 4 x is f − 1 ( x ) = 4 x . The x -intercept of a function is the y -intercept of its inverse. Since f ( x ) has no y -intercept, its inverse f − 1 ( x ) has no x -intercept. This is another reason why f ( x ) does not have a y -intercept.
Finding the X-intercept The x -intercept of a function occurs where the graph of the function intersects the x -axis. This happens when f ( x ) = 0 . So, we need to solve the equation lo g 4 x = 0 . We can rewrite this equation in exponential form as 4 0 = x . Since any non-zero number raised to the power of 0 is 1, we have x = 1 . Therefore, the x -intercept of f ( x ) = lo g 4 x is 1.
Final Answer In summary, f ( x ) = lo g 4 x does not have a y -intercept because there is no power of 4 that equals 0, and because its inverse does not have an x -intercept. The x -intercept of f ( x ) = lo g 4 x is 1.
Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and modeling population growth or decay. Understanding the intercepts of logarithmic functions helps us interpret these models. For example, in finance, the logarithm is used to calculate the time it takes for an investment to double at a fixed interest rate. Knowing the x-intercept helps determine the initial investment needed to start the growth process.
