How to find the x-intercept of a log function?
Understand the Problem
The question is asking how to determine the x-intercept of a logarithmic function. The x-intercept is the point where the function crosses the x-axis, which occurs when the function's value equals zero. To find the x-intercept of a log function, we set the function equal to zero and solve for x.
Answer
The x-intercept is $x = 1$.
Answer for screen readers
The x-intercept of the logarithmic function is $x = 1$.
Steps to Solve
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Set the logarithmic function equal to zero To find the x-intercept, set your logarithmic function, for example, $f(x) = \log_b(x)$, equal to zero: $$ \log_b(x) = 0 $$
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Rewrite in exponential form Next, rewrite the equation in its exponential form to eliminate the logarithm. Recall that if $\log_b(a) = c$, then $a = b^c$. Here: $$ x = b^0 $$
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Simplify the equation Since any number raised to the power of 0 is 1, we can simplify the equation to: $$ x = 1 $$
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State the x-intercept Conclude that the x-intercept of the logarithmic function $f(x) = \log_b(x)$ is: $$ (1, 0) $$
The x-intercept of the logarithmic function is $x = 1$.
More Information
The x-intercept shows where the function crosses the x-axis, meaning the logarithm of 1 is always 0, regardless of the base. This property holds for all logarithmic functions.
Tips
- Forgetting the base: Make sure to recognize the base of the logarithm when applying properties.
- Assuming negative inputs: Logarithmic functions are only defined for positive inputs, so remember that values for $x$ must be greater than 0.
