Algebra Examples

$10 - 2^{3 x + 1} = 5$

Step 1

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

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Step 1.1

Subtract $10$ from both sides of the equation.

$- 2^{3 x + 1} = 5 - 10$

Step 1.2

Subtract $10$ from $5$ .

$- 2^{3 x + 1} = - 5$

Step 2

Divide each term in $- 2^{3 x + 1} = - 5$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- 2^{3 x + 1} = - 5$ by $- 1$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.2

Divide $2^{3 x + 1}$ by $1$ .

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $- 5$ by $- 1$ .

$2^{3 x + 1} = 5$

Step 3

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

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

Step 4

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

$(3 x + 1) \ln (2) = \ln (5)$

Step 5

Simplify the left side.

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Step 5.1

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

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Step 5.1.1

Apply the distributive property.

$3 x \ln (2) + 1 \ln (2) = \ln (5)$

Step 5.1.2

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

$3 x \ln (2) + \ln (2) = \ln (5)$

Step 6

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

$3 x \ln (2) + \ln (2) - \ln (5) = 0$

Step 7

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

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

Step 8

Subtract $\ln (\frac{2}{5})$ from both sides of the equation.

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

Step 9

Divide each term in $3 x \ln (2) = - \ln (\frac{2}{5})$ by $3 \ln (2)$ and simplify.

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Step 9.1

Divide each term in $3 x \ln (2) = - \ln (\frac{2}{5})$ by $3 \ln (2)$ .

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

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $3$ .

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Step 9.2.1.1

Cancel the common factor.

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

Step 9.2.1.2

Rewrite the expression.

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

Step 9.2.2

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

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Step 9.2.2.1

Cancel the common factor.

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

Step 9.2.2.2

Divide $x$ by $1$ .

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

Step 9.3

Simplify the right side.

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Step 9.3.1

Move the negative in front of the fraction.

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

Step 10

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.44064269 \dots$