Finite Math Examples

$6 ({1.02}^{5 x + 1}) = 9$

Step 1

Divide each term in $6 \cdot {1.02}^{5 x + 1} = 9$ by $6$ and simplify.

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Step 1.1

Divide each term in $6 \cdot {1.02}^{5 x + 1} = 9$ by $6$ .

$\frac{6 \cdot {1.02}^{5 x + 1}}{6} = \frac{9}{6}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $6$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{6 \cdot {1.02}^{5 x + 1}}{6} = \frac{9}{6}$

Step 1.2.1.2

Divide ${1.02}^{5 x + 1}$ by $1$ .

${1.02}^{5 x + 1} = \frac{9}{6}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $9$ and $6$ .

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Step 1.3.1.1

Factor $3$ out of $9$ .

${1.02}^{5 x + 1} = \frac{3 (3)}{6}$

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $3$ out of $6$ .

${1.02}^{5 x + 1} = \frac{3 \cdot 3}{3 \cdot 2}$

Step 1.3.1.2.2

Cancel the common factor.

${1.02}^{5 x + 1} = \frac{3 \cdot 3}{3 \cdot 2}$

Step 1.3.1.2.3

Rewrite the expression.

${1.02}^{5 x + 1} = \frac{3}{2}$

Step 2

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

$\ln ({1.02}^{5 x + 1}) = \ln (\frac{3}{2})$

Step 3

Expand $\ln ({1.02}^{5 x + 1})$ by moving $5 x + 1$ outside the logarithm.

$(5 x + 1) \ln (1.02) = \ln (\frac{3}{2})$

Step 4

Simplify the left side.

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Step 4.1

Simplify $(5 x + 1) \ln (1.02)$ .

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Step 4.1.1

Apply the distributive property.

$5 x \ln (1.02) + 1 \ln (1.02) = \ln (\frac{3}{2})$

Step 4.1.2

Multiply $\ln (1.02)$ by $1$ .

$5 x \ln (1.02) + \ln (1.02) = \ln (\frac{3}{2})$

Step 5

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

$5 x \ln (1.02) + \ln (1.02) - \ln (\frac{3}{2}) = 0$

Step 6

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

$5 x \ln (1.02) + \ln (\frac{1.02}{\frac{3}{2}}) = 0$

Step 7

Simplify each term.

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Step 7.1

Multiply the numerator by the reciprocal of the denominator.

$5 x \ln (1.02) + \ln (1.02 (\frac{2}{3})) = 0$

Step 7.2

Multiply $1.02 (\frac{2}{3})$ .

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Step 7.2.1

Combine $1.02$ and $\frac{2}{3}$ .

$5 x \ln (1.02) + \ln (\frac{1.02 \cdot 2}{3}) = 0$

Step 7.2.2

Multiply $1.02$ by $2$ .

$5 x \ln (1.02) + \ln (\frac{2.04}{3}) = 0$

Step 7.3

Divide $2.04$ by $3$ .

$5 x \ln (1.02) + \ln (0.68) = 0$

Step 8

Subtract $\ln (0.68)$ from both sides of the equation.

$5 x \ln (1.02) = - \ln (0.68)$

Step 9

Divide each term in $5 x \ln (1.02) = - \ln (0.68)$ by $5 \ln (1.02)$ and simplify.

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Step 9.1

Divide each term in $5 x \ln (1.02) = - \ln (0.68)$ by $5 \ln (1.02)$ .

$\frac{5 x \ln (1.02)}{5 \ln (1.02)} = \frac{- \ln (0.68)}{5 \ln (1.02)}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $5$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{5 x \ln (1.02)}{5 \ln (1.02)} = \frac{- \ln (0.68)}{5 \ln (1.02)}$

Step 9.2.1.2

Rewrite the expression.

$\frac{x \ln (1.02)}{\ln (1.02)} = \frac{- \ln (0.68)}{5 \ln (1.02)}$

Step 9.2.2

Cancel the common factor of $\ln (1.02)$ .

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Step 9.2.2.1

Cancel the common factor.

$\frac{x \ln (1.02)}{\ln (1.02)} = \frac{- \ln (0.68)}{5 \ln (1.02)}$

Step 9.2.2.2

Divide $x$ by $1$ .

$x = \frac{- \ln (0.68)}{5 \ln (1.02)}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (0.68)}{5 \ln (1.02)}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (0.68)}{5 \ln (1.02)}$

Decimal Form:

$x = 3.89506377 \dots$