Finite Math Examples

$3^{2 x + 1} - 50 = 100$

Step 1

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

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Step 1.1

Add $50$ to both sides of the equation.

$3^{2 x + 1} = 100 + 50$

Step 1.2

Add $100$ and $50$ .

$3^{2 x + 1} = 150$

Step 2

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

$\ln (3^{2 x + 1}) = \ln (150)$

Step 3

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

$(2 x + 1) \ln (3) = \ln (150)$

Step 4

Simplify the left side.

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Step 4.1

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

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Step 4.1.1

Apply the distributive property.

$2 x \ln (3) + 1 \ln (3) = \ln (150)$

Step 4.1.2

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

$2 x \ln (3) + \ln (3) = \ln (150)$

Step 5

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

$2 x \ln (3) + \ln (3) - \ln (150) = 0$

Step 6

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

$2 x \ln (3) + \ln (\frac{3}{150}) = 0$

Step 7

Cancel the common factor of $3$ and $150$ .

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Step 7.1

Factor $3$ out of $3$ .

$2 x \ln (3) + \ln (\frac{3 (1)}{150}) = 0$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $3$ out of $150$ .

$2 x \ln (3) + \ln (\frac{3 \cdot 1}{3 \cdot 50}) = 0$

Step 7.2.2

Cancel the common factor.

$2 x \ln (3) + \ln (\frac{3 \cdot 1}{3 \cdot 50}) = 0$

Step 7.2.3

Rewrite the expression.

$2 x \ln (3) + \ln (\frac{1}{50}) = 0$

Step 8

Subtract $\ln (\frac{1}{50})$ from both sides of the equation.

$2 x \ln (3) = - \ln (\frac{1}{50})$

Step 9

Divide each term in $2 x \ln (3) = - \ln (\frac{1}{50})$ by $2 \ln (3)$ and simplify.

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Step 9.1

Divide each term in $2 x \ln (3) = - \ln (\frac{1}{50})$ by $2 \ln (3)$ .

$\frac{2 x \ln (3)}{2 \ln (3)} = \frac{- \ln (\frac{1}{50})}{2 \ln (3)}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $2$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{2 x \ln (3)}{2 \ln (3)} = \frac{- \ln (\frac{1}{50})}{2 \ln (3)}$

Step 9.2.1.2

Rewrite the expression.

$\frac{x \ln (3)}{\ln (3)} = \frac{- \ln (\frac{1}{50})}{2 \ln (3)}$

Step 9.2.2

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

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Step 9.2.2.1

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{- \ln (\frac{1}{50})}{2 \ln (3)}$

Step 9.2.2.2

Divide $x$ by $1$ .

$x = \frac{- \ln (\frac{1}{50})}{2 \ln (3)}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{1}{50})}{2 \ln (3)}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (\frac{1}{50})}{2 \ln (3)}$

Decimal Form:

$x = 1.78043839 \dots$