Algebra Examples

$f (x) = \frac{3}{2} \cdot {(\frac{4}{3})}^{x} - 1$

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 $x$ .

$0 = \frac{3}{2} \cdot {(\frac{4}{3})}^{x} - 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\frac{3}{2} \cdot {(\frac{4}{3})}^{x} - 1 = 0$ .

$\frac{3}{2} \cdot {(\frac{4}{3})}^{x} - 1 = 0$

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Apply the product rule to $\frac{4}{3}$ .

$\frac{3}{2} \cdot \frac{4^{x}}{3^{x}} - 1 = 0$

Step 1.2.2.2

Combine.

$\frac{3 \cdot 4^{x}}{2 \cdot 3^{x}} - 1 = 0$

Step 1.2.3

Add $1$ to both sides of the equation.

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

Step 1.2.4

Multiply both sides by $2 \cdot 3^{x}$ .

$\frac{3 \cdot 4^{x}}{2 \cdot 3^{x}} (2 \cdot 3^{x}) = 1 (2 \cdot 3^{x})$

Step 1.2.5

Simplify.

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Step 1.2.5.1

Simplify the left side.

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Step 1.2.5.1.1

Simplify $\frac{3 \cdot 4^{x}}{2 \cdot 3^{x}} (2 \cdot 3^{x})$ .

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Step 1.2.5.1.1.1

Rewrite using the commutative property of multiplication.

$2 \frac{3 \cdot 4^{x}}{2 \cdot 3^{x}} \cdot 3^{x} = 1 (2 \cdot 3^{x})$

Step 1.2.5.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.5.1.1.2.1

Factor $2$ out of $2 \cdot 3^{x}$ .

$2 \frac{3 \cdot 4^{x}}{2 (3^{x})} \cdot 3^{x} = 1 (2 \cdot 3^{x})$

Step 1.2.5.1.1.2.2

Cancel the common factor.

$2 \frac{3 \cdot 4^{x}}{2 \cdot 3^{x}} \cdot 3^{x} = 1 (2 \cdot 3^{x})$

Step 1.2.5.1.1.2.3

Rewrite the expression.

$\frac{3 \cdot 4^{x}}{3^{x}} \cdot 3^{x} = 1 (2 \cdot 3^{x})$

Step 1.2.5.1.1.3

Cancel the common factor of $3^{x}$ .

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Step 1.2.5.1.1.3.1

Cancel the common factor.

$\frac{3 \cdot 4^{x}}{3^{x}} \cdot 3^{x} = 1 (2 \cdot 3^{x})$

Step 1.2.5.1.1.3.2

Rewrite the expression.

$3 \cdot 4^{x} = 1 (2 \cdot 3^{x})$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Multiply $2 \cdot 3^{x}$ by $1$ .

$3 \cdot 4^{x} = 2 \cdot 3^{x}$

Step 1.2.6

Solve for $x$ .

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Step 1.2.6.1

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

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

Step 1.2.6.2

Expand the left side.

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Step 1.2.6.2.1

Rewrite $\ln (3 \cdot 4^{x})$ as $\ln (3) + \ln (4^{x})$ .

$\ln (3) + \ln (4^{x}) = \ln (2 \cdot 3^{x})$

Step 1.2.6.2.2

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

$\ln (3) + x \ln (4) = \ln (2 \cdot 3^{x})$

Step 1.2.6.3

Expand the right side.

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Step 1.2.6.3.1

Rewrite $\ln (2 \cdot 3^{x})$ as $\ln (2) + \ln (3^{x})$ .

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

Step 1.2.6.3.2

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

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

Step 1.2.6.4

Reorder $\ln (3)$ and $x \ln (4)$ .

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

Step 1.2.6.5

Reorder $\ln (2)$ and $x \ln (3)$ .

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

Step 1.2.6.6

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

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

Step 1.2.6.7

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

$x \ln (4) - x \ln (3) + \ln (\frac{3}{2}) = 0$

Step 1.2.6.8

Subtract $\ln (\frac{3}{2})$ from both sides of the equation.

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

Step 1.2.6.9

Factor $x$ out of $x \ln (4) - x \ln (3)$ .

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Step 1.2.6.9.1

Factor $x$ out of $x \ln (4)$ .

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

Step 1.2.6.9.2

Factor $x$ out of $- x \ln (3)$ .

$x (\ln (4)) + x (- 1 \ln (3)) = - \ln (\frac{3}{2})$

Step 1.2.6.9.3

Factor $x$ out of $x (\ln (4)) + x (- 1 \ln (3))$ .

$x (\ln (4) - 1 \ln (3)) = - \ln (\frac{3}{2})$

Step 1.2.6.10

Rewrite $- 1 \ln (3)$ as $- \ln (3)$ .

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

Step 1.2.6.11

Divide each term in $x (\ln (4) - \ln (3)) = - \ln (\frac{3}{2})$ by $\ln (4) - \ln (3)$ and simplify.

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Step 1.2.6.11.1

Divide each term in $x (\ln (4) - \ln (3)) = - \ln (\frac{3}{2})$ by $\ln (4) - \ln (3)$ .

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

Step 1.2.6.11.2

Simplify the left side.

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Step 1.2.6.11.2.1

Cancel the common factor of $\ln (4) - \ln (3)$ .

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Step 1.2.6.11.2.1.1

Cancel the common factor.

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

Step 1.2.6.11.2.1.2

Divide $x$ by $1$ .

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

Step 1.2.6.11.3

Simplify the right side.

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Step 1.2.6.11.3.1

Move the negative in front of the fraction.

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

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

$y = \frac{3}{2} \cdot {(\frac{4}{3})}^{0} - 1$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{3}{2} \cdot {(\frac{4}{3})}^{0} - 1$

Step 2.2.2

Simplify $\frac{3}{2} \cdot {(\frac{4}{3})}^{0} - 1$ .

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Apply the product rule to $\frac{4}{3}$ .

$y = \frac{3}{2} \cdot \frac{4^{0}}{3^{0}} - 1$

Step 2.2.2.1.2

Combine.

$y = \frac{3 \cdot 4^{0}}{2 \cdot 3^{0}} - 1$

Step 2.2.2.1.3

Cancel the common factor of $3$ and $3^{0}$ .

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Step 2.2.2.1.3.1

Factor $3$ out of $3 \cdot 4^{0}$ .

$y = \frac{3 (4^{0})}{2 \cdot 3^{0}} - 1$

Step 2.2.2.1.3.2

Cancel the common factors.

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Step 2.2.2.1.3.2.1

Factor $3$ out of $2 \cdot 3^{0}$ .

$y = \frac{3 (4^{0})}{3 (2 \cdot 3^{- 1})} - 1$

Step 2.2.2.1.3.2.2

Cancel the common factor.

$y = \frac{3 \cdot 4^{0}}{3 (2 \cdot 3^{- 1})} - 1$

Step 2.2.2.1.3.2.3

Rewrite the expression.

$y = \frac{4^{0}}{2 \cdot 3^{- 1}} - 1$

Step 2.2.2.1.4

Move $3^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$y = \frac{4^{0} \cdot 3}{2} - 1$

Step 2.2.2.1.5

Simplify the numerator.

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Step 2.2.2.1.5.1

Anything raised to $0$ is $1$ .

$y = \frac{1 \cdot 3}{2} - 1$

Step 2.2.2.1.5.2

Multiply $3$ by $1$ .

$y = \frac{3}{2} - 1$

Step 2.2.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{3}{2} - 1 \cdot \frac{2}{2}$

Step 2.2.2.3

Combine $- 1$ and $\frac{2}{2}$ .

$y = \frac{3}{2} + \frac{- 1 \cdot 2}{2}$

Step 2.2.2.4

Combine the numerators over the common denominator.

$y = \frac{3 - 1 \cdot 2}{2}$

Step 2.2.2.5

Simplify the numerator.

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Step 2.2.2.5.1

Multiply $- 1$ by $2$ .

$y = \frac{3 - 2}{2}$

Step 2.2.2.5.2

Subtract $2$ from $3$ .

$y = \frac{1}{2}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{1}{2})$

Step 3

List the intersections.

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

y-intercept(s): $(0, \frac{1}{2})$

Step 4