Precalculus Examples

$y = 2^{x + 2} - 5$

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 $x$ .

$0 = 2^{x + 2} - 5$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $2^{x + 2} - 5 = 0$ .

$2^{x + 2} - 5 = 0$

Step 1.2.2

Add $5$ to both sides of the equation.

$2^{x + 2} = 5$

Step 1.2.3

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

$\ln (2^{x + 2}) = \ln (5)$

Step 1.2.4

Expand $\ln (2^{x + 2})$ by moving $x + 2$ outside the logarithm.

$(x + 2) \ln (2) = \ln (5)$

Step 1.2.5

Simplify the left side.

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Step 1.2.5.1

Apply the distributive property.

$x \ln (2) + 2 \ln (2) = \ln (5)$

Step 1.2.6

Move all the terms containing a logarithm to the left side of the equation.

$x \ln (2) + 2 \ln (2) - \ln (5) = 0$

Step 1.2.7

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.7.1

Subtract $2 \ln (2)$ from both sides of the equation.

$x \ln (2) - \ln (5) = - 2 \ln (2)$

Step 1.2.7.2

Add $\ln (5)$ to both sides of the equation.

$x \ln (2) = - 2 \ln (2) + \ln (5)$

Step 1.2.8

Divide each term in $x \ln (2) = - 2 \ln (2) + \ln (5)$ by $\ln (2)$ and simplify.

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Step 1.2.8.1

Divide each term in $x \ln (2) = - 2 \ln (2) + \ln (5)$ by $\ln (2)$ .

$\frac{x \ln (2)}{\ln (2)} = \frac{- 2 \ln (2)}{\ln (2)} + \frac{\ln (5)}{\ln (2)}$

Step 1.2.8.2

Simplify the left side.

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Step 1.2.8.2.1

Cancel the common factor of $\ln (2)$ .

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Step 1.2.8.2.1.1

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{- 2 \ln (2)}{\ln (2)} + \frac{\ln (5)}{\ln (2)}$

Step 1.2.8.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2 \ln (2)}{\ln (2)} + \frac{\ln (5)}{\ln (2)}$

Step 1.2.8.3

Simplify the right side.

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Step 1.2.8.3.1

Cancel the common factor of $\ln (2)$ .

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Step 1.2.8.3.1.1

Cancel the common factor.

$x = \frac{- 2 \ln (2)}{\ln (2)} + \frac{\ln (5)}{\ln (2)}$

Step 1.2.8.3.1.2

Divide $- 2$ by $1$ .

$x = - 2 + \frac{\ln (5)}{\ln (2)}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 2 + \frac{\ln (5)}{\ln (2)}, 0)$

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 $y$ .

$y = 2^{(0) + 2} - 5$

Step 2.2

Simplify $2^{(0) + 2} - 5$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Add $0$ and $2$ .

$y = 2^{2} - 5$

Step 2.2.1.2

Raise $2$ to the power of $2$ .

$y = 4 - 5$

Step 2.2.2

Subtract $5$ from $4$ .

$y = - 1$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(- 2 + \frac{\ln (5)}{\ln (2)}, 0)$

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

Step 4