Trigonometry Examples

$\log_{19} (3) + \log_{19} (x + 7) = \log_{19} (8 x + 6)$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{19} (3 (x + 7)) = \log_{19} (8 x + 6)$

Step 1.2

Apply the distributive property.

$\log_{19} (3 x + 3 \cdot 7) = \log_{19} (8 x + 6)$

Step 1.3

Multiply $3$ by $7$ .

$\log_{19} (3 x + 21) = \log_{19} (8 x + 6)$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$3 x + 21 = 8 x + 6$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms containing $x$ to the left side of the equation.

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Step 3.1.1

Subtract $8 x$ from both sides of the equation.

$3 x + 21 - 8 x = 6$

Step 3.1.2

Subtract $8 x$ from $3 x$ .

$- 5 x + 21 = 6$

Step 3.2

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

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Step 3.2.1

Subtract $21$ from both sides of the equation.

$- 5 x = 6 - 21$

Step 3.2.2

Subtract $21$ from $6$ .

$- 5 x = - 15$

Step 3.3

Divide each term in $- 5 x = - 15$ by $- 5$ and simplify.

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Step 3.3.1

Divide each term in $- 5 x = - 15$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{- 15}{- 5}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 5$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{- 15}{- 5}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 15}{- 5}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $- 15$ by $- 5$ .

$x = 3$