Trigonometry Examples

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

Step 1

Move all terms not containing $\log (2 x + 5)$ to the right side of the equation.

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Step 1.1

Subtract $3$ from both sides of the equation.

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

Step 1.2

Subtract $3$ from $2$ .

$\log (2 x + 5) = - 1$

Step 2

Rewrite $\log (2 x + 5) = - 1$ 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 $b^{y} = x$ .

$10^{- 1} = 2 x + 5$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $2 x + 5 = 10^{- 1}$ .

$2 x + 5 = 10^{- 1}$

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

Subtract $5$ from both sides of the equation.

$2 x = \frac{1}{10} - 5$

Step 3.3.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$2 x = \frac{1}{10} - 5 \cdot \frac{10}{10}$

Step 3.3.3

Combine $- 5$ and $\frac{10}{10}$ .

$2 x = \frac{1}{10} + \frac{- 5 \cdot 10}{10}$

Step 3.3.4

Combine the numerators over the common denominator.

$2 x = \frac{1 - 5 \cdot 10}{10}$

Step 3.3.5

Simplify the numerator.

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Step 3.3.5.1

Multiply $- 5$ by $10$ .

$2 x = \frac{1 - 50}{10}$

Step 3.3.5.2

Subtract $50$ from $1$ .

$2 x = \frac{- 49}{10}$

Step 3.3.6

Move the negative in front of the fraction.

$2 x = - \frac{49}{10}$

Step 3.4

Divide each term in $2 x = - \frac{49}{10}$ by $2$ and simplify.

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Step 3.4.1

Divide each term in $2 x = - \frac{49}{10}$ by $2$ .

$\frac{2 x}{2} = \frac{- \frac{49}{10}}{2}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- \frac{49}{10}}{2}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{49}{10}}{2}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{49}{10} \cdot \frac{1}{2}$

Step 3.4.3.2

Multiply $- \frac{49}{10} \cdot \frac{1}{2}$ .

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Step 3.4.3.2.1

Multiply $\frac{1}{2}$ by $\frac{49}{10}$ .

$x = - \frac{49}{2 \cdot 10}$

Step 3.4.3.2.2

Multiply $2$ by $10$ .

$x = - \frac{49}{20}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{49}{20}$

Decimal Form:

$x = - 2.45$

Mixed Number Form:

$x = - 2 \frac{9}{20}$