Find the Vertical and Horizontal Intercept(s) f(x)=4^(x+1)-10

$f (x) = 4^{(x + 1)} - 10$

Step 1

Find the x-intercepts.

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

$0 = 4^{x + 1} - 10$

Solve the equation.

Rewrite the equation as $4^{x + 1} - 10 = 0$ .

$4^{x + 1} - 10 = 0$

Add $10$ to both sides of the equation.

$4^{x + 1} = 10$

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

$\ln (4^{x + 1}) = \ln (10)$

Expand $\ln (4^{x + 1})$ by moving $x + 1$ outside the logarithm.

$(x + 1) \ln (4) = \ln (10)$

Simplify the left side.

Simplify $(x + 1) \ln (4)$ .

Apply the distributive property.

$x \ln (4) + 1 \ln (4) = \ln (10)$

Multiply $\ln (4)$ by $1$ .

$x \ln (4) + \ln (4) = \ln (10)$

Move all the terms containing a logarithm to the left side of the equation.

$x \ln (4) + \ln (4) - \ln (10) = 0$

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$x \ln (4) + \ln (\frac{4}{10}) = 0$

Cancel the common factor of $4$ and $10$ .

Factor $2$ out of $4$ .

$x \ln (4) + \ln (\frac{2 (2)}{10}) = 0$

Cancel the common factors.

Factor $2$ out of $10$ .

$x \ln (4) + \ln (\frac{2 \cdot 2}{2 \cdot 5}) = 0$

Cancel the common factor.

$x \ln (4) + \ln (\frac{2 \cdot 2}{2 \cdot 5}) = 0$

Rewrite the expression.

$x \ln (4) + \ln (\frac{2}{5}) = 0$

Subtract $\ln (\frac{2}{5})$ from both sides of the equation.

$x \ln (4) = - \ln (\frac{2}{5})$

Divide each term in $x \ln (4) = - \ln (\frac{2}{5})$ by $\ln (4)$ and simplify.

Divide each term in $x \ln (4) = - \ln (\frac{2}{5})$ by $\ln (4)$ .

$\frac{x \ln (4)}{\ln (4)} = \frac{- \ln (\frac{2}{5})}{\ln (4)}$

Simplify the left side.

Cancel the common factor of $\ln (4)$ .

Cancel the common factor.

$\frac{x \ln (4)}{\ln (4)} = \frac{- \ln (\frac{2}{5})}{\ln (4)}$

Divide $x$ by $1$ .

$x = \frac{- \ln (\frac{2}{5})}{\ln (4)}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{2}{5})}{\ln (4)}$

x-intercept(s) in point form.

x-intercept(s): $(- \frac{\ln (\frac{2}{5})}{\ln (4)}, 0)$

Step 2

Find the y-intercepts.

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

$y = 4^{(0) + 1} - 10$

Simplify $4^{(0) + 1} - 10$ .

Simplify each term.

Add $0$ and $1$ .

$y = 4^{1} - 10$

Evaluate the exponent.

$y = 4 - 10$

Subtract $10$ from $4$ .

$y = - 6$

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(- \frac{\ln (\frac{2}{5})}{\ln (4)}, 0)$

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

Step 4

Find the Vertical and Horizontal Intercept(s) f(x)=4^(x+1)-10

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