Precalculus Examples

$5 (2^{3 - x}) - 13 = 100$

Step 1

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

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Step 1.1

Add $13$ to both sides of the equation.

$5 \cdot 2^{3 - x} = 100 + 13$

Step 1.2

Add $100$ and $13$ .

$5 \cdot 2^{3 - x} = 113$

Step 2

Divide each term in $5 \cdot 2^{3 - x} = 113$ by $5$ and simplify.

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Step 2.1

Divide each term in $5 \cdot 2^{3 - x} = 113$ by $5$ .

$\frac{5 \cdot 2^{3 - x}}{5} = \frac{113}{5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $5$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{5 \cdot 2^{3 - x}}{5} = \frac{113}{5}$

Step 2.2.1.2

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

$2^{3 - x} = \frac{113}{5}$

Step 3

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

$\ln (2^{3 - x}) = \ln (\frac{113}{5})$

Step 4

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

$(3 - x) \ln (2) = \ln (\frac{113}{5})$

Step 5

Simplify the left side.

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Step 5.1

Apply the distributive property.

$3 \ln (2) - x \ln (2) = \ln (\frac{113}{5})$

Step 6

Reorder $3 \ln (2)$ and $- x \ln (2)$ .

$- x \ln (2) + 3 \ln (2) = \ln (\frac{113}{5})$

Step 7

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

$- x \ln (2) + 3 \ln (2) - \ln (\frac{113}{5}) = 0$

Step 8

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

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Step 8.1

Subtract $3 \ln (2)$ from both sides of the equation.

$- x \ln (2) - \ln (\frac{113}{5}) = - 3 \ln (2)$

Step 8.2

Add $\ln (\frac{113}{5})$ to both sides of the equation.

$- x \ln (2) = - 3 \ln (2) + \ln (\frac{113}{5})$

Step 9

Divide each term in $- x \ln (2) = - 3 \ln (2) + \ln (\frac{113}{5})$ by $- \ln (2)$ and simplify.

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Step 9.1

Divide each term in $- x \ln (2) = - 3 \ln (2) + \ln (\frac{113}{5})$ by $- \ln (2)$ .

$\frac{- x \ln (2)}{- \ln (2)} = \frac{- 3 \ln (2)}{- \ln (2)} + \frac{\ln (\frac{113}{5})}{- \ln (2)}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Dividing two negative values results in a positive value.

$\frac{x \ln (2)}{\ln (2)} = \frac{- 3 \ln (2)}{- \ln (2)} + \frac{\ln (\frac{113}{5})}{- \ln (2)}$

Step 9.2.2

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

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Step 9.2.2.1

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{- 3 \ln (2)}{- \ln (2)} + \frac{\ln (\frac{113}{5})}{- \ln (2)}$

Step 9.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 3 \ln (2)}{- \ln (2)} + \frac{\ln (\frac{113}{5})}{- \ln (2)}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Simplify each term.

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Step 9.3.1.1

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

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Step 9.3.1.1.1

Cancel the common factor.

$x = \frac{- 3 \ln (2)}{- \ln (2)} + \frac{\ln (\frac{113}{5})}{- \ln (2)}$

Step 9.3.1.1.2

Rewrite the expression.

$x = \frac{- 3}{- 1} + \frac{\ln (\frac{113}{5})}{- \ln (2)}$

Step 9.3.1.1.3

Move the negative one from the denominator of $\frac{- 3}{- 1}$ .

$x = - 1 \cdot - 3 + \frac{\ln (\frac{113}{5})}{- \ln (2)}$

Step 9.3.1.2

Rewrite $- 1 \cdot - 3$ as $- - 3$ .

$x = - - 3 + \frac{\ln (\frac{113}{5})}{- \ln (2)}$

Step 9.3.1.3

Multiply $- 1$ by $- 3$ .

$x = 3 + \frac{\ln (\frac{113}{5})}{- \ln (2)}$

Step 9.3.1.4

Move the negative in front of the fraction.

$x = 3 - \frac{\ln (\frac{113}{5})}{\ln (2)}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = 3 - \frac{\ln (\frac{113}{5})}{\ln (2)}$

Decimal Form:

$x = - 1.49825086 \dots$