Algebra Examples

$n (x) = - 4^{5 - 2 x} + 3$

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 $x$ .

$0 = - 4^{5 - 2 x} + 3$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 4^{5 - 2 x} + 3 = 0$ .

$- 4^{5 - 2 x} + 3 = 0$

Step 1.2.2

Subtract $3$ from both sides of the equation.

$- 4^{5 - 2 x} = - 3$

Step 1.2.3

Divide each term in $- 4^{5 - 2 x} = - 3$ by $- 1$ and simplify.

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Step 1.2.3.1

Divide each term in $- 4^{5 - 2 x} = - 3$ by $- 1$ .

$\frac{- 4^{5 - 2 x}}{- 1} = \frac{- 3}{- 1}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{4^{5 - 2 x}}{1} = \frac{- 3}{- 1}$

Step 1.2.3.2.2

Divide $4^{5 - 2 x}$ by $1$ .

$4^{5 - 2 x} = \frac{- 3}{- 1}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $- 3$ by $- 1$ .

$4^{5 - 2 x} = 3$

Step 1.2.4

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

$\ln (4^{5 - 2 x}) = \ln (3)$

Step 1.2.5

Expand $\ln (4^{5 - 2 x})$ by moving $5 - 2 x$ outside the logarithm.

$(5 - 2 x) \ln (4) = \ln (3)$

Step 1.2.6

Simplify the left side.

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Step 1.2.6.1

Apply the distributive property.

$5 \ln (4) - 2 x \ln (4) = \ln (3)$

Step 1.2.7

Reorder $5 \ln (4)$ and $- 2 x \ln (4)$ .

$- 2 x \ln (4) + 5 \ln (4) = \ln (3)$

Step 1.2.8

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

$- 2 x \ln (4) + 5 \ln (4) - \ln (3) = 0$

Step 1.2.9

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.9.1

Subtract $5 \ln (4)$ from both sides of the equation.

$- 2 x \ln (4) - \ln (3) = - 5 \ln (4)$

Step 1.2.9.2

Add $\ln (3)$ to both sides of the equation.

$- 2 x \ln (4) = - 5 \ln (4) + \ln (3)$

Step 1.2.10

Divide each term in $- 2 x \ln (4) = - 5 \ln (4) + \ln (3)$ by $- 2 \ln (4)$ and simplify.

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Step 1.2.10.1

Divide each term in $- 2 x \ln (4) = - 5 \ln (4) + \ln (3)$ by $- 2 \ln (4)$ .

$\frac{- 2 x \ln (4)}{- 2 \ln (4)} = \frac{- 5 \ln (4)}{- 2 \ln (4)} + \frac{\ln (3)}{- 2 \ln (4)}$

Step 1.2.10.2

Simplify the left side.

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Step 1.2.10.2.1

Cancel the common factor of $- 2$ .

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Step 1.2.10.2.1.1

Cancel the common factor.

$\frac{- 2 x \ln (4)}{- 2 \ln (4)} = \frac{- 5 \ln (4)}{- 2 \ln (4)} + \frac{\ln (3)}{- 2 \ln (4)}$

Step 1.2.10.2.1.2

Rewrite the expression.

$\frac{x \ln (4)}{\ln (4)} = \frac{- 5 \ln (4)}{- 2 \ln (4)} + \frac{\ln (3)}{- 2 \ln (4)}$

Step 1.2.10.2.2

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

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Step 1.2.10.2.2.1

Cancel the common factor.

$\frac{x \ln (4)}{\ln (4)} = \frac{- 5 \ln (4)}{- 2 \ln (4)} + \frac{\ln (3)}{- 2 \ln (4)}$

Step 1.2.10.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 5 \ln (4)}{- 2 \ln (4)} + \frac{\ln (3)}{- 2 \ln (4)}$

Step 1.2.10.3

Simplify the right side.

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Step 1.2.10.3.1

Simplify each term.

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Step 1.2.10.3.1.1

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

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Step 1.2.10.3.1.1.1

Cancel the common factor.

$x = \frac{- 5 \ln (4)}{- 2 \ln (4)} + \frac{\ln (3)}{- 2 \ln (4)}$

Step 1.2.10.3.1.1.2

Rewrite the expression.

$x = \frac{- 5}{- 2} + \frac{\ln (3)}{- 2 \ln (4)}$

Step 1.2.10.3.1.2

Dividing two negative values results in a positive value.

$x = \frac{5}{2} + \frac{\ln (3)}{- 2 \ln (4)}$

Step 1.2.10.3.1.3

Move the negative in front of the fraction.

$x = \frac{5}{2} - \frac{\ln (3)}{2 \ln (4)}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{5}{2} - \frac{\ln (3)}{2 \ln (4)}, 0)$

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 $y$ .

$y = - 4^{5 - 2 \cdot 0} + 3$

Step 2.2

Simplify $- 4^{5 - 2 \cdot 0} + 3$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $- 2$ by $0$ .

$y = - 4^{5 + 0} + 3$

Step 2.2.1.2

Add $5$ and $0$ .

$y = - 4^{5} + 3$

Step 2.2.1.3

Raise $4$ to the power of $5$ .

$y = - 1 \cdot 1024 + 3$

Step 2.2.1.4

Multiply $- 1$ by $1024$ .

$y = - 1024 + 3$

Step 2.2.2

Add $- 1024$ and $3$ .

$y = - 1021$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 1021)$

Step 3

List the intersections.

x-intercept(s): $(\frac{5}{2} - \frac{\ln (3)}{2 \ln (4)}, 0)$

y-intercept(s): $(0, - 1021)$

Step 4