Algebra Examples

$f (x) = 7^{x} - 4$

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 $x$ .

$0 = 7^{x} - 4$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $7^{x} - 4 = 0$ .

$7^{x} - 4 = 0$

Step 1.2.2

Add $4$ to both sides of the equation.

$7^{x} = 4$

Step 1.2.3

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

$\ln (7^{x}) = \ln (4)$

Step 1.2.4

Expand $\ln (7^{x})$ by moving $x$ outside the logarithm.

$x \ln (7) = \ln (4)$

Step 1.2.5

Divide each term in $x \ln (7) = \ln (4)$ by $\ln (7)$ and simplify.

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Step 1.2.5.1

Divide each term in $x \ln (7) = \ln (4)$ by $\ln (7)$ .

$\frac{x \ln (7)}{\ln (7)} = \frac{\ln (4)}{\ln (7)}$

Step 1.2.5.2

Simplify the left side.

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Step 1.2.5.2.1

Cancel the common factor of $\ln (7)$ .

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Step 1.2.5.2.1.1

Cancel the common factor.

$\frac{x \ln (7)}{\ln (7)} = \frac{\ln (4)}{\ln (7)}$

Step 1.2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (4)}{\ln (7)}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{\ln (4)}{\ln (7)}, 0)$

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 $y$ .

$y = 7^{0} - 4$

Step 2.2

Simplify $7^{0} - 4$ .

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Step 2.2.1

Anything raised to $0$ is $1$ .

$y = 1 - 4$

Step 2.2.2

Subtract $4$ from $1$ .

$y = - 3$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 3)$

Step 3

List the intersections.

x-intercept(s): $(\frac{\ln (4)}{\ln (7)}, 0)$

y-intercept(s): $(0, - 3)$

Step 4