Algebra Examples

$f (x) = e^{x} - 5$

Step 1

Find the x-intercepts.

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Step 1.1

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

$0 = e^{x} - 5$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $e^{x} - 5 = 0$ .

$e^{x} - 5 = 0$

Step 1.2.2

Add $5$ to both sides of the equation.

$e^{x} = 5$

Step 1.2.3

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{x}) = \ln (5)$

Step 1.2.4

Expand the left side.

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Step 1.2.4.1

Expand $\ln (e^{x})$ by moving $x$ outside the logarithm.

$x \ln (e) = \ln (5)$

Step 1.2.4.2

The natural logarithm of $e$ is $1$ .

$x \cdot 1 = \ln (5)$

Step 1.2.4.3

Multiply $x$ by $1$ .

$x = \ln (5)$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\ln (5), 0)$

Step 2

Find the y-intercepts.

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Step 2.1

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

$y = e^{0} - 5$

Step 2.2

Simplify $e^{0} - 5$ .

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Step 2.2.1

Anything raised to $0$ is $1$ .

$y = 1 - 5$

Step 2.2.2

Subtract $5$ from $1$ .

$y = - 4$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 4)$

Step 3

List the intersections.

x-intercept(s): $(\ln (5), 0)$

y-intercept(s): $(0, - 4)$

Step 4