Algebra Examples

$2.3 = 2^{\frac{x}{9} + 1} + 1$

Step 1

Rewrite the equation as $2^{\frac{x}{9} + 1} + 1 = 2.3$ .

$2^{\frac{x}{9} + 1} + 1 = 2.3$

Step 2

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

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Step 2.1

Subtract $1$ from both sides of the equation.

$2^{\frac{x}{9} + 1} = 2.3 - 1$

Step 2.2

Subtract $1$ from $2.3$ .

$2^{\frac{x}{9} + 1} = 1.3$

Step 3

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

$\ln (2^{\frac{x}{9} + 1}) = \ln (1.3)$

Step 4

Expand $\ln (2^{\frac{x}{9} + 1})$ by moving $\frac{x}{9} + 1$ outside the logarithm.

$(\frac{x}{9} + 1) \ln (2) = \ln (1.3)$

Step 5

Simplify the left side.

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Step 5.1

Simplify $(\frac{x}{9} + 1) \ln (2)$ .

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Step 5.1.1

Apply the distributive property.

$\frac{x}{9} \ln (2) + 1 \ln (2) = \ln (1.3)$

Step 5.1.2

Combine $\frac{x}{9}$ and $\ln (2)$ .

$\frac{x \ln (2)}{9} + 1 \ln (2) = \ln (1.3)$

Step 5.1.3

Multiply $\ln (2)$ by $1$ .

$\frac{x \ln (2)}{9} + \ln (2) = \ln (1.3)$

Step 6

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

$\frac{x \ln (2)}{9} + \ln (2) - \ln (1.3) = 0$

Step 7

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

$\frac{x \ln (2)}{9} + \ln (\frac{2}{1.3}) = 0$

Step 8

Divide $2$ by $1.3$ .

$\frac{x \ln (2)}{9} + \ln (1.\overline{538461}) = 0$

Step 9

Subtract $\ln (1.\overline{538461})$ from both sides of the equation.

$\frac{x \ln (2)}{9} = - \ln (1.\overline{538461})$

Step 10

Multiply both sides of the equation by $9$ .

$9 \frac{x \ln (2)}{9} = 9 (- \ln (1.\overline{538461}))$

Step 11

Simplify both sides of the equation.

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Step 11.1

Simplify the left side.

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Step 11.1.1

Cancel the common factor of $9$ .

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Step 11.1.1.1

Cancel the common factor.

$9 \frac{x \ln (2)}{9} = 9 (- \ln (1.\overline{538461}))$

Step 11.1.1.2

Rewrite the expression.

$x \ln (2) = 9 (- \ln (1.\overline{538461}))$

Step 11.2

Simplify the right side.

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Step 11.2.1

Multiply $- 1$ by $9$ .

$x \ln (2) = - 9 \ln (1.\overline{538461})$

Step 12

Divide each term in $x \ln (2) = - 9 \ln (1.\overline{538461})$ by $\ln (2)$ and simplify.

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Step 12.1

Divide each term in $x \ln (2) = - 9 \ln (1.\overline{538461})$ by $\ln (2)$ .

$\frac{x \ln (2)}{\ln (2)} = \frac{- 9 \ln (1.\overline{538461})}{\ln (2)}$

Step 12.2

Simplify the left side.

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Step 12.2.1

Cancel the common factor of $\ln (2)$ .

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Step 12.2.1.1

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{- 9 \ln (1.\overline{538461})}{\ln (2)}$

Step 12.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 9 \ln (1.\overline{538461})}{\ln (2)}$

Step 12.3

Simplify the right side.

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Step 12.3.1

Move the negative in front of the fraction.

$x = - \frac{9 \ln (1.\overline{538461})}{\ln (2)}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{9 \ln (1.\overline{538461})}{\ln (2)}$

Decimal Form:

$x = - 5.59339539 \dots$