Precalculus Examples

$2^{3 - x} = 565$

Step 1

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

$\ln (2^{3 - x}) = \ln (565)$

Step 2

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

$(3 - x) \ln (2) = \ln (565)$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

$3 \ln (2) - x \ln (2) = \ln (565)$

Step 4

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

$- x \ln (2) + 3 \ln (2) = \ln (565)$

Step 5

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

$- x \ln (2) + 3 \ln (2) - \ln (565) = 0$

Step 6

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

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Step 6.1

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

$- x \ln (2) - \ln (565) = - 3 \ln (2)$

Step 6.2

Add $\ln (565)$ to both sides of the equation.

$- x \ln (2) = - 3 \ln (2) + \ln (565)$

Step 7

Divide each term in $- x \ln (2) = - 3 \ln (2) + \ln (565)$ by $- \ln (2)$ and simplify.

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Step 7.1

Divide each term in $- x \ln (2) = - 3 \ln (2) + \ln (565)$ by $- \ln (2)$ .

$\frac{- x \ln (2)}{- \ln (2)} = \frac{- 3 \ln (2)}{- \ln (2)} + \frac{\ln (565)}{- \ln (2)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Dividing two negative values results in a positive value.

$\frac{x \ln (2)}{\ln (2)} = \frac{- 3 \ln (2)}{- \ln (2)} + \frac{\ln (565)}{- \ln (2)}$

Step 7.2.2

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

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Step 7.2.2.1

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{- 3 \ln (2)}{- \ln (2)} + \frac{\ln (565)}{- \ln (2)}$

Step 7.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 3 \ln (2)}{- \ln (2)} + \frac{\ln (565)}{- \ln (2)}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

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

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Step 7.3.1.1.1

Cancel the common factor.

$x = \frac{- 3 \ln (2)}{- \ln (2)} + \frac{\ln (565)}{- \ln (2)}$

Step 7.3.1.1.2

Rewrite the expression.

$x = \frac{- 3}{- 1} + \frac{\ln (565)}{- \ln (2)}$

Step 7.3.1.1.3

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

$x = - 1 \cdot - 3 + \frac{\ln (565)}{- \ln (2)}$

Step 7.3.1.2

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

$x = - - 3 + \frac{\ln (565)}{- \ln (2)}$

Step 7.3.1.3

Multiply $- 1$ by $- 3$ .

$x = 3 + \frac{\ln (565)}{- \ln (2)}$

Step 7.3.1.4

Move the negative in front of the fraction.

$x = 3 - \frac{\ln (565)}{\ln (2)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = 3 - \frac{\ln (565)}{\ln (2)}$

Decimal Form:

$x = - 6.14210705 \dots$