Question

The graph of \(f(x)=\log _{a} x,\) where \(a>0\)and \(a \neq 1,\) has an \(x\) -intercept equal to 1 and no \(y\) -intercept.

Short answer

The graph intercepts the x-axis at 1 and has no y-intercept because \(\log_a{x}\) is undefined at \(x = 0\).

Step-by-step solution

Identify the function and its properties

Given the function is \(f(x) = \log_a{x}\), where the base \(a\) is greater than zero and not equal to one. It is noted that the graph has an \(x\)-intercept at 1 and no \(y\)-intercept.

Analyze the x-intercept condition

To find the \(x\)-intercept, set \(f(x) = 0\). That is \(\log_a{x} = 0\). This equation implies that \(a^0 = x\), hence \(x = 1\). This matches the given x-intercept condition at \(x = 1\).

Reason why there is no y-intercept

For the y-intercept, we set \(x = 0\) in the function \(f(x) = \log_a{x}\). However, the logarithmic function \(\log_a{x}\) is undefined for \(x = 0\). Thus, there is no y-intercept as the logarithm of zero is not defined.

Key concepts

x-intercept

The concept of an x-intercept is crucial in understanding every graph. It refers to the point where the graph crosses the x-axis. For the function given in the exercise, which is \(f(x) = \text{{log}}_a{x}\), an x-intercept occurs where the output value is zero.

To find this, we solve the equation \text{{log}}_a{x} = 0. By definition, logarithms ask the question: \