Precalculus Examples

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

Step 1

Rewrite $\log (3 x + 5) = - 2$ 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 $b^{y} = x$ .

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

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $3 x + 5 = 10^{- 2}$ .

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

Step 2.2

Simplify $10^{- 2}$ .

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Step 2.2.1

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

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

Step 2.2.2

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

$3 x + 5 = \frac{1}{100}$

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 $5$ from both sides of the equation.

$3 x = \frac{1}{100} - 5$

Step 2.3.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{100}{100}$ .

$3 x = \frac{1}{100} - 5 \cdot \frac{100}{100}$

Step 2.3.3

Combine $- 5$ and $\frac{100}{100}$ .

$3 x = \frac{1}{100} + \frac{- 5 \cdot 100}{100}$

Step 2.3.4

Combine the numerators over the common denominator.

$3 x = \frac{1 - 5 \cdot 100}{100}$

Step 2.3.5

Simplify the numerator.

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Step 2.3.5.1

Multiply $- 5$ by $100$ .

$3 x = \frac{1 - 500}{100}$

Step 2.3.5.2

Subtract $500$ from $1$ .

$3 x = \frac{- 499}{100}$

Step 2.3.6

Move the negative in front of the fraction.

$3 x = - \frac{499}{100}$

Step 2.4

Divide each term in $3 x = - \frac{499}{100}$ by $3$ and simplify.

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Step 2.4.1

Divide each term in $3 x = - \frac{499}{100}$ by $3$ .

$\frac{3 x}{3} = \frac{- \frac{499}{100}}{3}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $3$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- \frac{499}{100}}{3}$

Step 2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{499}{100}}{3}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{499}{100} \cdot \frac{1}{3}$

Step 2.4.3.2

Multiply $- \frac{499}{100} \cdot \frac{1}{3}$ .

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Step 2.4.3.2.1

Multiply $\frac{1}{3}$ by $\frac{499}{100}$ .

$x = - \frac{499}{3 \cdot 100}$

Step 2.4.3.2.2

Multiply $3$ by $100$ .

$x = - \frac{499}{300}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{499}{300}$

Decimal Form:

$x = - 1.66\overline{3}$

Mixed Number Form:

$x = - 1 \frac{199}{300}$