Finite Math Examples

${(\frac{2}{3})}^{3 x - 1} > 1$

Step 1

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

$\ln ({(\frac{2}{3})}^{3 x - 1}) > \ln (1)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln ({(\frac{2}{3})}^{3 x - 1})$ by moving $3 x - 1$ outside the logarithm.

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

Step 2.2

Rewrite $\ln (\frac{2}{3})$ as $\ln (2) - \ln (3)$ .

$(3 x - 1) (\ln (2) - \ln (3)) > \ln (1)$

Step 3

Simplify the left side.

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Step 3.1

Simplify $(3 x - 1) (\ln (2) - \ln (3))$ .

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Step 3.1.1

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

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

Step 3.1.2

Apply the distributive property.

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

Step 3.1.3

Rewrite $- 1 \ln (\frac{2}{3})$ as $- \ln (\frac{2}{3})$ .

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

Step 4

Simplify the right side.

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Step 4.1

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

$3 x \ln (\frac{2}{3}) - \ln (\frac{2}{3}) > 0$

Step 5

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

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $3$ .

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Rewrite the expression.

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

Step 7.2.2

Cancel the common factor of $\ln (\frac{2}{3})$ .

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Step 7.2.2.1

Cancel the common factor.

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

Step 7.2.2.2

Divide $x$ by $1$ .

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

Step 7.3

Simplify the right side.

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Step 7.3.1

Cancel the common factor of $\ln (\frac{2}{3})$ .

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Step 7.3.1.1

Cancel the common factor.

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

Step 7.3.1.2

Rewrite the expression.

$x = \frac{1}{3}$

Step 8

The solution consists of all of the true intervals.

$x < \frac{1}{3}$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$x < \frac{1}{3}$

Interval Notation:

$(- \infty, \frac{1}{3})$

Step 10