Question

How do you find the vertical asymptote of a logarithmic function?

Answer 1

Hint: Horizontal asymptote is the value that any function in x-variable approaches infinity x approaches infinity and as x approaches negative infinity. Vertical asymptotes occur wherever the value of function approaches infinity or negative infinity. Keeping this information in mind, we shall proceed to find the vertical asymptote of a logarithmic function.

Complete step-by-step solution:
Vertical asymptote is basically an invisible vertical line that the graph of a function approaches and gets really close to but never actually touches. The point that the graph of a function never touches is the point where it is undefined.
We know that the logarithmic function is not defined at $x=0$. This is the equation of y-axis as well. Therefore, we understand that the graph of logarithmic function never touches the y-axis but it approaches negative infinity at the y-axis.
The graph of $\log x$ function is given below.

Even though the function seems to be approaching the y-axis but it never touches it in real. Therefore, the vertical asymptote of a logarithmic function is the y-axis.

Note: As we approach from the left, a vertical asymptote looks like the graph is approaching a vertical up to infinity but it never intersects with any point on the x-axis as it is not defined at that point. Similarly, as we approach from the right, the graph again does not intersect at any point and approaches a vertical up to infinity as it is undefined at that same point.