Precalculus Examples

$\log_{3} (2 - x) = 3$

Step 1

Rewrite $\log_{3} (2 - x) = 3$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$3^{3} = 2 - x$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $2 - x = 3^{3}$ .

$2 - x = 3^{3}$

Step 2.2

Raise $3$ to the power of $3$ .

$2 - x = 27$

Step 2.3

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

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Step 2.3.1

Subtract $2$ from both sides of the equation.

$- x = 27 - 2$

Step 2.3.2

Subtract $2$ from $27$ .

$- x = 25$

Step 2.4

Divide each term in $- x = 25$ by $- 1$ and simplify.

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Step 2.4.1

Divide each term in $- x = 25$ by $- 1$ .

$\frac{- x}{- 1} = \frac{25}{- 1}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{25}{- 1}$

Step 2.4.2.2

Divide $x$ by $1$ .

$x = \frac{25}{- 1}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Divide $25$ by $- 1$ .

$x = - 25$