Trigonometry Examples

$\log_{6} (5) (2 y - 3) = \log_{6} (25)$

Step 1

Divide each term in $\log_{6} (5) (2 y - 3) = \log_{6} (25)$ by $\log_{6} (5)$ and simplify.

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Step 1.1

Divide each term in $\log_{6} (5) (2 y - 3) = \log_{6} (25)$ by $\log_{6} (5)$ .

$\frac{\log_{6} (5) (2 y - 3)}{\log_{6} (5)} = \frac{\log_{6} (25)}{\log_{6} (5)}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $\log_{6} (5)$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{\log_{6} (5) (2 y - 3)}{\log_{6} (5)} = \frac{\log_{6} (25)}{\log_{6} (5)}$

Step 1.2.1.2

Divide $2 y - 3$ by $1$ .

$2 y - 3 = \frac{\log_{6} (25)}{\log_{6} (5)}$

Step 2

Add $3$ to both sides of the equation.

$2 y = \frac{\log_{6} (25)}{\log_{6} (5)} + 3$

Step 3

Divide each term in $2 y = \frac{\log_{6} (25)}{\log_{6} (5)} + 3$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 y = \frac{\log_{6} (25)}{\log_{6} (5)} + 3$ by $2$ .

$\frac{2 y}{2} = \frac{\frac{\log_{6} (25)}{\log_{6} (5)}}{2} + \frac{3}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{\frac{\log_{6} (25)}{\log_{6} (5)}}{2} + \frac{3}{2}$

Step 3.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{\log_{6} (25)}{\log_{6} (5)}}{2} + \frac{3}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{\log_{6} (25)}{\log_{6} (5)} \cdot \frac{1}{2} + \frac{3}{2}$

Step 3.3.1.2

Multiply $\frac{\log_{6} (25)}{\log_{6} (5)}$ by $\frac{1}{2}$ .

$y = \frac{\log_{6} (25)}{\log_{6} (5) \cdot 2} + \frac{3}{2}$

Step 3.3.1.3

Move $2$ to the left of $\log_{6} (5)$ .

$y = \frac{\log_{6} (25)}{2 \log_{6} (5)} + \frac{3}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$y = \frac{\log_{6} (25)}{2 \log_{6} (5)} + \frac{3}{2}$

Decimal Form:

$y = 2.5$