Trigonometry Examples

$2 \log (50) = 3 \log (25) + \log (x - 2)$

Step 1

Rewrite the equation as $3 \log (25) + \log (x - 2) = 2 \log (50)$ .

$3 \log (25) + \log (x - 2) = 2 \log (50)$

Step 2

Move all the terms containing a logarithm to the left side of the equation.

$3 \log (25) + \log (x - 2) - 2 \log (50) = 0$

Step 3

Simplify the left side.

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Step 3.1

Simplify $3 \log (25) + \log (x - 2) - 2 \log (50)$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Simplify $3 \log (25)$ by moving $3$ inside the logarithm.

$\log (25^{3}) + \log (x - 2) - 2 \log (50) = 0$

Step 3.1.1.2

Raise $25$ to the power of $3$ .

$\log (15625) + \log (x - 2) - 2 \log (50) = 0$

Step 3.1.1.3

Simplify $- 2 \log (50)$ by moving $2$ inside the logarithm.

$\log (15625) + \log (x - 2) - \log (50^{2}) = 0$

Step 3.1.1.4

Raise $50$ to the power of $2$ .

$\log (15625) + \log (x - 2) - \log (2500) = 0$

Step 3.1.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (15625 (x - 2)) - \log (2500) = 0$

Step 3.1.3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{15625 (x - 2)}{2500}) = 0$

Step 3.1.4

Cancel the common factor of $15625$ and $2500$ .

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Step 3.1.4.1

Factor $625$ out of $15625 (x - 2)$ .

$\log (\frac{625 (25 (x - 2))}{2500}) = 0$

Step 3.1.4.2

Cancel the common factors.

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Step 3.1.4.2.1

Factor $625$ out of $2500$ .

$\log (\frac{625 (25 (x - 2))}{625 (4)}) = 0$

Step 3.1.4.2.2

Cancel the common factor.

$\log (\frac{625 (25 (x - 2))}{625 \cdot 4}) = 0$

Step 3.1.4.2.3

Rewrite the expression.

$\log (\frac{25 (x - 2)}{4}) = 0$

Step 4

Rewrite $\log (\frac{25 (x - 2)}{4}) = 0$ 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 $b^{y} = x$ .

$10^{0} = \frac{25 (x - 2)}{4}$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $\frac{25 (x - 2)}{4} = 10^{0}$ .

$\frac{25 (x - 2)}{4} = 10^{0}$

Step 5.2

Multiply both sides of the equation by $\frac{4}{25}$ .

$\frac{4}{25} \cdot \frac{25 (x - 2)}{4} = \frac{4}{25} \cdot 10^{0}$

Step 5.3

Simplify both sides of the equation.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Simplify $\frac{4}{25} \cdot \frac{25 (x - 2)}{4}$ .

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Step 5.3.1.1.1

Cancel the common factor of $4$ .

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Step 5.3.1.1.1.1

Cancel the common factor.

$\frac{4}{25} \cdot \frac{25 (x - 2)}{4} = \frac{4}{25} \cdot 10^{0}$

Step 5.3.1.1.1.2

Rewrite the expression.

$\frac{1}{25} (25 (x - 2)) = \frac{4}{25} \cdot 10^{0}$

Step 5.3.1.1.2

Cancel the common factor of $25$ .

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Step 5.3.1.1.2.1

Cancel the common factor.

$\frac{1}{25} (25 (x - 2)) = \frac{4}{25} \cdot 10^{0}$

Step 5.3.1.1.2.2

Rewrite the expression.

$x - 2 = \frac{4}{25} \cdot 10^{0}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Simplify $\frac{4}{25} \cdot 10^{0}$ .

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Step 5.3.2.1.1

Anything raised to $0$ is $1$ .

$x - 2 = \frac{4}{25} \cdot 1$

Step 5.3.2.1.2

Multiply $\frac{4}{25}$ by $1$ .

$x - 2 = \frac{4}{25}$

Step 5.4

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

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Step 5.4.1

Add $2$ to both sides of the equation.

$x = \frac{4}{25} + 2$

Step 5.4.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$x = \frac{4}{25} + 2 \cdot \frac{25}{25}$

Step 5.4.3

Combine $2$ and $\frac{25}{25}$ .

$x = \frac{4}{25} + \frac{2 \cdot 25}{25}$

Step 5.4.4

Combine the numerators over the common denominator.

$x = \frac{4 + 2 \cdot 25}{25}$

Step 5.4.5

Simplify the numerator.

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Step 5.4.5.1

Multiply $2$ by $25$ .

$x = \frac{4 + 50}{25}$

Step 5.4.5.2

Add $4$ and $50$ .

$x = \frac{54}{25}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{54}{25}$

Decimal Form:

$x = 2.16$

Mixed Number Form:

$x = 2 \frac{4}{25}$