Trigonometry Examples

$\log_{5} (2 x) - 5 = - 4$

Step 1

Move all terms not containing $\log_{5} (2 x)$ to the right side of the equation.

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Step 1.1

Add $5$ to both sides of the equation.

$\log_{5} (2 x) = - 4 + 5$

Step 1.2

Add $- 4$ and $5$ .

$\log_{5} (2 x) = 1$

Step 2

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

$5^{1} = 2 x$

Step 3

Solve for $x$ .

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Step 3.1

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

$2 x = 5$

Step 3.2

Divide each term in $2 x = 5^{1}$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 x = 5^{1}$ by $2$ .

$\frac{2 x}{2} = \frac{5^{1}}{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{5^{1}}{2}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5^{1}}{2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Evaluate the exponent.

$x = \frac{5}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{2}$

Decimal Form:

$x = 2.5$

Mixed Number Form:

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