Precalculus Examples

$\log_{3} (5 x) = \log_{3} (x) + \log_{3} (2 x + 1)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $\log_{3} (x) + \log_{3} (2 x + 1)$ .

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Step 1.1.1

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

$\log_{3} (5 x) = \log_{3} (x (2 x + 1))$

Step 1.1.2

Apply the distributive property.

$\log_{3} (5 x) = \log_{3} (x (2 x) + x \cdot 1)$

Step 1.1.3

Simplify the expression.

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Step 1.1.3.1

Rewrite using the commutative property of multiplication.

$\log_{3} (5 x) = \log_{3} (2 x \cdot x + x \cdot 1)$

Step 1.1.3.2

Multiply $x$ by $1$ .

$\log_{3} (5 x) = \log_{3} (2 x \cdot x + x)$

Step 1.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.4.1

Move $x$ .

$\log_{3} (5 x) = \log_{3} (2 (x \cdot x) + x)$

Step 1.1.4.2

Multiply $x$ by $x$ .

$\log_{3} (5 x) = \log_{3} (2 x^{2} + x)$

Step 2

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

$5 x = 2 x^{2} + x$

Step 3

Solve for $x$ .

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Step 3.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$2 x^{2} + x = 5 x$

Step 3.2

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

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Step 3.2.1

Subtract $5 x$ from both sides of the equation.

$2 x^{2} + x - 5 x = 0$

Step 3.2.2

Subtract $5 x$ from $x$ .

$2 x^{2} - 4 x = 0$

Step 3.3

Factor $2 x$ out of $2 x^{2} - 4 x$ .

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Step 3.3.1

Factor $2 x$ out of $2 x^{2}$ .

$2 x (x) - 4 x = 0$

Step 3.3.2

Factor $2 x$ out of $- 4 x$ .

$2 x (x) + 2 x (- 2) = 0$

Step 3.3.3

Factor $2 x$ out of $2 x (x) + 2 x (- 2)$ .

$2 x (x - 2) = 0$

Step 3.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 2 = 0$

Step 3.5

Set $x$ equal to $0$ .

$x = 0$

Step 3.6

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 3.6.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 3.6.2

Add $2$ to both sides of the equation.

$x = 2$

Step 3.7

The final solution is all the values that make $2 x (x - 2) = 0$ true.

$x = 0, 2$

Step 4

Exclude the solutions that do not make $\log_{3} (5 x) = \log_{3} (x) + \log_{3} (2 x + 1)$ true.

$x = 2$