Algebra Examples

$y + 3 = 2^{x}$

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 $x$ .

$(0) + 3 = 2^{x}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $2^{x} = (0) + 3$ .

$2^{x} = (0) + 3$

Step 1.2.2

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

$\ln (2^{x}) = \ln (3)$

Step 1.2.3

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

$x \ln (2) = \ln (3)$

Step 1.2.4

Divide each term in $x \ln (2) = \ln (3)$ by $\ln (2)$ and simplify.

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Step 1.2.4.1

Divide each term in $x \ln (2) = \ln (3)$ by $\ln (2)$ .

$\frac{x \ln (2)}{\ln (2)} = \frac{\ln (3)}{\ln (2)}$

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

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

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Step 1.2.4.2.1.1

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{\ln (3)}{\ln (2)}$

Step 1.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (3)}{\ln (2)}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{\ln (3)}{\ln (2)}, 0)$

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 $y$ .

$y + 3 = 2^{0}$

Step 2.2

Solve the equation.

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Step 2.2.1

Anything raised to $0$ is $1$ .

$y + 3 = 1$

Step 2.2.2

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

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Step 2.2.2.1

Subtract $3$ from both sides of the equation.

$y = 1 - 3$

Step 2.2.2.2

Subtract $3$ from $1$ .

$y = - 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 2)$

Step 3

List the intersections.

x-intercept(s): $(\frac{\ln (3)}{\ln (2)}, 0)$

y-intercept(s): $(0, - 2)$

Step 4