Algebra Examples

$y = 3^{1 - x} - 4$

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 $x$ .

$0 = 3^{1 - x} - 4$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $3^{1 - x} - 4 = 0$ .

$3^{1 - x} - 4 = 0$

Step 1.2.2

Add $4$ to both sides of the equation.

$3^{1 - x} = 4$

Step 1.2.3

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

$\ln (3^{1 - x}) = \ln (4)$

Step 1.2.4

Expand $\ln (3^{1 - x})$ by moving $1 - x$ outside the logarithm.

$(1 - x) \ln (3) = \ln (4)$

Step 1.2.5

Simplify the left side.

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Step 1.2.5.1

Simplify $(1 - x) \ln (3)$ .

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Step 1.2.5.1.1

Apply the distributive property.

$1 \ln (3) - x \ln (3) = \ln (4)$

Step 1.2.5.1.2

Multiply $\ln (3)$ by $1$ .

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

Step 1.2.6

Reorder $\ln (3)$ and $- x \ln (3)$ .

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

Step 1.2.7

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

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

Step 1.2.8

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

$- x \ln (3) + \ln (\frac{3}{4}) = 0$

Step 1.2.9

Subtract $\ln (\frac{3}{4})$ from both sides of the equation.

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

Step 1.2.10

Divide each term in $- x \ln (3) = - \ln (\frac{3}{4})$ by $- \ln (3)$ and simplify.

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Step 1.2.10.1

Divide each term in $- x \ln (3) = - \ln (\frac{3}{4})$ by $- \ln (3)$ .

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

Step 1.2.10.2

Simplify the left side.

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Step 1.2.10.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.10.2.2

Cancel the common factor of $\ln (3)$ .

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Step 1.2.10.2.2.1

Cancel the common factor.

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

Step 1.2.10.2.2.2

Divide $x$ by $1$ .

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

Step 1.2.10.3

Simplify the right side.

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Step 1.2.10.3.1

Dividing two negative values results in a positive value.

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

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

$y = 3^{1 - (0)} - 4$

Step 2.2

Simplify $3^{1 - (0)} - 4$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Subtract $0$ from $1$ .

$y = 3^{1} - 4$

Step 2.2.1.2

Evaluate the exponent.

$y = 3 - 4$

Step 2.2.2

Subtract $4$ from $3$ .

$y = - 1$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 1)$

Step 3

List the intersections.

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

y-intercept(s): $(0, - 1)$

Step 4