Precalculus Examples

$3 - \log (x + 2) = 5$

Step 1

Move all terms not containing $\log (x + 2)$ to the right side of the equation.

Tap for more steps...

Step 1.1

Subtract $3$ from both sides of the equation.

$- \log (x + 2) = 5 - 3$

Step 1.2

Subtract $3$ from $5$ .

$- \log (x + 2) = 2$

Step 2

Divide each term in $- \log (x + 2) = 2$ by $- 1$ and simplify.

Tap for more steps...

Step 2.1

Divide each term in $- \log (x + 2) = 2$ by $- 1$ .

$\frac{- \log (x + 2)}{- 1} = \frac{2}{- 1}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{\log (x + 2)}{1} = \frac{2}{- 1}$

Step 2.2.2

Divide $\log (x + 2)$ by $1$ .

$\log (x + 2) = \frac{2}{- 1}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Divide $2$ by $- 1$ .

$\log (x + 2) = - 2$

Step 3

Rewrite $\log (x + 2) = - 2$ 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 $b^{y} = x$ .

$10^{- 2} = x + 2$

Step 4

Solve for $x$ .

Tap for more steps...

Step 4.1

Rewrite the equation as $x + 2 = 10^{- 2}$ .

$x + 2 = 10^{- 2}$

Step 4.2

Simplify $10^{- 2}$ .

Tap for more steps...

Step 4.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x + 2 = \frac{1}{10^{2}}$

Step 4.2.2

Raise $10$ to the power of $2$ .

$x + 2 = \frac{1}{100}$

Step 4.3

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

Tap for more steps...

Step 4.3.1

Subtract $2$ from both sides of the equation.

$x = \frac{1}{100} - 2$

Step 4.3.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{100}{100}$ .

$x = \frac{1}{100} - 2 \cdot \frac{100}{100}$

Step 4.3.3

Combine $- 2$ and $\frac{100}{100}$ .

$x = \frac{1}{100} + \frac{- 2 \cdot 100}{100}$

Step 4.3.4

Combine the numerators over the common denominator.

$x = \frac{1 - 2 \cdot 100}{100}$

Step 4.3.5

Simplify the numerator.

Tap for more steps...

Step 4.3.5.1

Multiply $- 2$ by $100$ .

$x = \frac{1 - 200}{100}$

Step 4.3.5.2

Subtract $200$ from $1$ .

$x = \frac{- 199}{100}$

Step 4.3.6

Move the negative in front of the fraction.

$x = - \frac{199}{100}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{199}{100}$

Decimal Form:

$x = - 1.99$

Mixed Number Form:

$x = - 1 \frac{99}{100}$