Find the x and y Intercepts f(x)=100^(x-5)-1

$f (x) = 100^{x - 5} - 1$

Step 1

Find the x-intercepts.

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 100^{x - 5} - 1$

Solve the equation.

Rewrite the equation as $100^{x - 5} - 1 = 0$ .

$100^{x - 5} - 1 = 0$

Add $1$ to both sides of the equation.

$100^{x - 5} = 1$

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

$\ln (100^{x - 5}) = \ln (1)$

Expand $\ln (100^{x - 5})$ by moving $x - 5$ outside the logarithm.

$(x - 5) \ln (100) = \ln (1)$

Simplify the left side.

Apply the distributive property.

$x \ln (100) - 5 \ln (100) = \ln (1)$

Simplify the right side.

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

$x \ln (100) - 5 \ln (100) = 0$

Add $5 \ln (100)$ to both sides of the equation.

$x \ln (100) = 5 \ln (100)$

Divide each term in $x \ln (100) = 5 \ln (100)$ by $\ln (100)$ and simplify.

Divide each term in $x \ln (100) = 5 \ln (100)$ by $\ln (100)$ .

$\frac{x \ln (100)}{\ln (100)} = \frac{5 \ln (100)}{\ln (100)}$

Simplify the left side.

Cancel the common factor of $\ln (100)$ .

Cancel the common factor.

$\frac{x \ln (100)}{\ln (100)} = \frac{5 \ln (100)}{\ln (100)}$

Divide $x$ by $1$ .

$x = \frac{5 \ln (100)}{\ln (100)}$

Simplify the right side.

Cancel the common factor of $\ln (100)$ .

Cancel the common factor.

$x = \frac{5 \ln (100)}{\ln (100)}$

Divide $5$ by $1$ .

$x = 5$

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 100^{(0) - 5} - 1$

Simplify $100^{(0) - 5} - 1$ .

Simplify each term.

Subtract $5$ from $0$ .

$y = 100^{- 5} - 1$

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

$y = \frac{1}{100^{5}} - 1$

Raise $100$ to the power of $5$ .

$y = \frac{1}{10000000000} - 1$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{10000000000}{10000000000}$ .

$y = \frac{1}{10000000000} - 1 \cdot \frac{10000000000}{10000000000}$

Combine $- 1$ and $\frac{10000000000}{10000000000}$ .

$y = \frac{1}{10000000000} + \frac{- 1 \cdot 10000000000}{10000000000}$

Combine the numerators over the common denominator.

$y = \frac{1 - 1 \cdot 10000000000}{10000000000}$

Simplify the numerator.

Multiply $- 1$ by $10000000000$ .

$y = \frac{1 - 10000000000}{10000000000}$

Subtract $10000000000$ from $1$ .

$y = \frac{- 9999999999}{10000000000}$

Move the negative in front of the fraction.

$y = - \frac{9999999999}{10000000000}$

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{9999999999}{10000000000})$

Step 3

List the intersections.

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

y-intercept(s): $(0, - \frac{9999999999}{10000000000})$

Step 4

Find the x and y Intercepts f(x)=100^(x-5)-1

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