Trigonometry Examples

$- 9 + \log_{5} (- 8 x - 1) = - 5$

Step 1

Move all terms not containing $\log_{5} (- 8 x - 1)$ to the right side of the equation.

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Step 1.1

Add $9$ to both sides of the equation.

$\log_{5} (- 8 x - 1) = - 5 + 9$

Step 1.2

Add $- 5$ and $9$ .

$\log_{5} (- 8 x - 1) = 4$

Step 2

Rewrite $\log_{5} (- 8 x - 1) = 4$ 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 $b^{y} = x$ .

$5^{4} = - 8 x - 1$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $- 8 x - 1 = 5^{4}$ .

$- 8 x - 1 = 5^{4}$

Step 3.2

Raise $5$ to the power of $4$ .

$- 8 x - 1 = 625$

Step 3.3

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

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Step 3.3.1

Add $1$ to both sides of the equation.

$- 8 x = 625 + 1$

Step 3.3.2

Add $625$ and $1$ .

$- 8 x = 626$

Step 3.4

Divide each term in $- 8 x = 626$ by $- 8$ and simplify.

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Step 3.4.1

Divide each term in $- 8 x = 626$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{626}{- 8}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $- 8$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{626}{- 8}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{626}{- 8}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Cancel the common factor of $626$ and $- 8$ .

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Step 3.4.3.1.1

Factor $2$ out of $626$ .

$x = \frac{2 (313)}{- 8}$

Step 3.4.3.1.2

Cancel the common factors.

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Step 3.4.3.1.2.1

Factor $2$ out of $- 8$ .

$x = \frac{2 \cdot 313}{2 \cdot - 4}$

Step 3.4.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 313}{2 \cdot - 4}$

Step 3.4.3.1.2.3

Rewrite the expression.

$x = \frac{313}{- 4}$

Step 3.4.3.2

Move the negative in front of the fraction.

$x = - \frac{313}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{313}{4}$

Decimal Form:

$x = - 78.25$

Mixed Number Form:

$x = - 78 \frac{1}{4}$