Calculus Examples

$\log_{3} (x)$

Step 1

Write $\log_{3} (x)$ as an equation.

$y = \log_{3} (x)$

Step 2

Find the x-intercepts.

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To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \log_{3} (x)$

Solve the equation.

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Rewrite the equation as $\log_{3} (x) = 0$ .

$\log_{3} (x) = 0$

Rewrite $\log_{3} (x) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$3^{0} = x$

Solve for $x$ .

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Rewrite the equation as $x = 3^{0}$ .

$x = 3^{0}$

Anything raised to $0$ is $1$ .

$x = 1$

x-intercept(s) in point form.

x-intercept(s): $(1, 0)$

Step 3

Find the y-intercepts.

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To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \log_{3} (0)$

Solve the equation.

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The logarithm of zero is undefined.

$y = Undefined$

Remove parentheses.

$y = \log_{3} (0)$

The equation cannot be solved because it is undefined.

$Undefined$

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

y-intercept(s): $None$

Step 4

List the intersections.

x-intercept(s): $(1, 0)$

y-intercept(s): $None$

Step 5