Calculus Examples

$\ln (x) + 2$

Write $\ln (x) + 2$ as an equation.

$y = \ln (x) + 2$

Find the x-intercepts.

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

$0 = \ln (x) + 2$

Solve the equation.

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Rewrite the equation as $\ln (x) + 2 = 0$ .

$\ln (x) + 2 = 0$

Subtract $2$ from both sides of the equation.

$\ln (x) = - 2$

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (x)} = e^{- 2}$

Rewrite $\ln (x) = - 2$ 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 $b^{y} = x$ .

$e^{- 2} = x$

Solve for $x$ .

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Rewrite the equation as $x = e^{- 2}$ .

$x = e^{- 2}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x = \frac{1}{e^{2}}$

x-intercept(s) in point form.

x-intercept(s): $(\frac{1}{e^{2}}, 0)$

Find the y-intercepts.

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

$y = \ln (0) + 2$

Solve the equation.

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The natural logarithm of zero is undefined.

$y = Undefined$

Remove parentheses.

$y = \ln (0) + 2$

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): $None$

List the intersections.

x-intercept(s): $(\frac{1}{e^{2}}, 0)$

y-intercept(s): $None$