Finite Math Examples

$5 ({1.06}^{2 x + 1}) = 11$

Step 1

Divide each term in $5 \cdot {1.06}^{2 x + 1} = 11$ by $5$ and simplify.

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Step 1.1

Divide each term in $5 \cdot {1.06}^{2 x + 1} = 11$ by $5$ .

$\frac{5 \cdot {1.06}^{2 x + 1}}{5} = \frac{11}{5}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $5$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{5 \cdot {1.06}^{2 x + 1}}{5} = \frac{11}{5}$

Step 1.2.1.2

Divide ${1.06}^{2 x + 1}$ by $1$ .

${1.06}^{2 x + 1} = \frac{11}{5}$

Step 2

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

$\ln ({1.06}^{2 x + 1}) = \ln (\frac{11}{5})$

Step 3

Expand $\ln ({1.06}^{2 x + 1})$ by moving $2 x + 1$ outside the logarithm.

$(2 x + 1) \ln (1.06) = \ln (\frac{11}{5})$

Step 4

Simplify the left side.

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Step 4.1

Simplify $(2 x + 1) \ln (1.06)$ .

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Step 4.1.1

Apply the distributive property.

$2 x \ln (1.06) + 1 \ln (1.06) = \ln (\frac{11}{5})$

Step 4.1.2

Multiply $\ln (1.06)$ by $1$ .

$2 x \ln (1.06) + \ln (1.06) = \ln (\frac{11}{5})$

Step 5

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

$2 x \ln (1.06) + \ln (1.06) - \ln (\frac{11}{5}) = 0$

Step 6

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

$2 x \ln (1.06) + \ln (\frac{1.06}{\frac{11}{5}}) = 0$

Step 7

Simplify each term.

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Step 7.1

Multiply the numerator by the reciprocal of the denominator.

$2 x \ln (1.06) + \ln (1.06 (\frac{5}{11})) = 0$

Step 7.2

Multiply $1.06 (\frac{5}{11})$ .

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Step 7.2.1

Combine $1.06$ and $\frac{5}{11}$ .

$2 x \ln (1.06) + \ln (\frac{1.06 \cdot 5}{11}) = 0$

Step 7.2.2

Multiply $1.06$ by $5$ .

$2 x \ln (1.06) + \ln (\frac{5.3}{11}) = 0$

Step 7.3

Divide $5.3$ by $11$ .

$2 x \ln (1.06) + \ln (0.4\overline{81}) = 0$

Step 8

Subtract $\ln (0.4\overline{81})$ from both sides of the equation.

$2 x \ln (1.06) = - \ln (0.4\overline{81})$

Step 9

Divide each term in $2 x \ln (1.06) = - \ln (0.4\overline{81})$ by $2 \ln (1.06)$ and simplify.

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Step 9.1

Divide each term in $2 x \ln (1.06) = - \ln (0.4\overline{81})$ by $2 \ln (1.06)$ .

$\frac{2 x \ln (1.06)}{2 \ln (1.06)} = \frac{- \ln (0.4\overline{81})}{2 \ln (1.06)}$

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.

$\frac{2 x \ln (1.06)}{2 \ln (1.06)} = \frac{- \ln (0.4\overline{81})}{2 \ln (1.06)}$

Step 9.2.1.2

Rewrite the expression.

$\frac{x \ln (1.06)}{\ln (1.06)} = \frac{- \ln (0.4\overline{81})}{2 \ln (1.06)}$

Step 9.2.2

Cancel the common factor of $\ln (1.06)$ .

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Step 9.2.2.1

Cancel the common factor.

$\frac{x \ln (1.06)}{\ln (1.06)} = \frac{- \ln (0.4\overline{81})}{2 \ln (1.06)}$

Step 9.2.2.2

Divide $x$ by $1$ .

$x = \frac{- \ln (0.4\overline{81})}{2 \ln (1.06)}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (0.4\overline{81})}{2 \ln (1.06)}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (0.4\overline{81})}{2 \ln (1.06)}$

Decimal Form:

$x = 6.26567817 \dots$