Algebra Examples

$f (x) = 3^{x} - 7$

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 $x$ .

$0 = 3^{x} - 7$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $3^{x} - 7 = 0$ .

$3^{x} - 7 = 0$

Step 1.2.2

Add $7$ to both sides of the equation.

$3^{x} = 7$

Step 1.2.3

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

$\ln (3^{x}) = \ln (7)$

Step 1.2.4

Expand $\ln (3^{x})$ by moving $x$ outside the logarithm.

$x \ln (3) = \ln (7)$

Step 1.2.5

Divide each term in $x \ln (3) = \ln (7)$ by $\ln (3)$ and simplify.

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Step 1.2.5.1

Divide each term in $x \ln (3) = \ln (7)$ by $\ln (3)$ .

$\frac{x \ln (3)}{\ln (3)} = \frac{\ln (7)}{\ln (3)}$

Step 1.2.5.2

Simplify the left side.

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Step 1.2.5.2.1

Cancel the common factor of $\ln (3)$ .

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Step 1.2.5.2.1.1

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{\ln (7)}{\ln (3)}$

Step 1.2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (7)}{\ln (3)}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{\ln (7)}{\ln (3)}, 0)$

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 $y$ .

$y = 3^{0} - 7$

Step 2.2

Simplify $3^{0} - 7$ .

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Step 2.2.1

Anything raised to $0$ is $1$ .

$y = 1 - 7$

Step 2.2.2

Subtract $7$ from $1$ .

$y = - 6$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 6)$

Step 3

List the intersections.

x-intercept(s): $(\frac{\ln (7)}{\ln (3)}, 0)$

y-intercept(s): $(0, - 6)$

Step 4