Algebra Examples

$f (x) = 5^{x - 5} - 1$

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 $x$ .

$0 = 5^{x - 5} - 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $5^{x - 5} - 1 = 0$ .

$5^{x - 5} - 1 = 0$

Step 1.2.2

Add $1$ to both sides of the equation.

$5^{x - 5} = 1$

Step 1.2.3

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

$\ln (5^{x - 5}) = \ln (1)$

Step 1.2.4

Expand $\ln (5^{x - 5})$ by moving $x - 5$ outside the logarithm.

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

Step 1.2.5

Simplify the left side.

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Step 1.2.5.1

Apply the distributive property.

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

Step 1.2.6

Simplify the right side.

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Step 1.2.6.1

The natural logarithm of $1$ is $0$ .

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

Step 1.2.7

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

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

Step 1.2.8

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

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Step 1.2.8.1

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

$\frac{x \ln (5)}{\ln (5)} = \frac{5 \ln (5)}{\ln (5)}$

Step 1.2.8.2

Simplify the left side.

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Step 1.2.8.2.1

Cancel the common factor of $\ln (5)$ .

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Step 1.2.8.2.1.1

Cancel the common factor.

$\frac{x \ln (5)}{\ln (5)} = \frac{5 \ln (5)}{\ln (5)}$

Step 1.2.8.2.1.2

Divide $x$ by $1$ .

$x = \frac{5 \ln (5)}{\ln (5)}$

Step 1.2.8.3

Simplify the right side.

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Step 1.2.8.3.1

Cancel the common factor of $\ln (5)$ .

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Step 1.2.8.3.1.1

Cancel the common factor.

$x = \frac{5 \ln (5)}{\ln (5)}$

Step 1.2.8.3.1.2

Divide $5$ by $1$ .

$x = 5$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

$y = 5^{(0) - 5} - 1$

Step 2.2

Simplify $5^{(0) - 5} - 1$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Subtract $5$ from $0$ .

$y = 5^{- 5} - 1$

Step 2.2.1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$y = \frac{1}{5^{5}} - 1$

Step 2.2.1.3

Raise $5$ to the power of $5$ .

$y = \frac{1}{3125} - 1$

Step 2.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3125}{3125}$ .

$y = \frac{1}{3125} - 1 \cdot \frac{3125}{3125}$

Step 2.2.3

Combine $- 1$ and $\frac{3125}{3125}$ .

$y = \frac{1}{3125} + \frac{- 1 \cdot 3125}{3125}$

Step 2.2.4

Combine the numerators over the common denominator.

$y = \frac{1 - 1 \cdot 3125}{3125}$

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Multiply $- 1$ by $3125$ .

$y = \frac{1 - 3125}{3125}$

Step 2.2.5.2

Subtract $3125$ from $1$ .

$y = \frac{- 3124}{3125}$

Step 2.2.6

Move the negative in front of the fraction.

$y = - \frac{3124}{3125}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{3124}{3125})$

Step 3

List the intersections.

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

y-intercept(s): $(0, - \frac{3124}{3125})$

Step 4