Precalculus Examples

$\ln (x + 3) = \ln (x) + \ln (3)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $\ln (x) + \ln (3)$ .

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Step 1.1.1

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

$\ln (x + 3) = \ln (x \cdot 3)$

Step 1.1.2

Move $3$ to the left of $x$ .

$\ln (x + 3) = \ln (3 x)$

Step 2

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

$x + 3 = 3 x$

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 $3 x$ from both sides of the equation.

$x + 3 - 3 x = 0$

Step 3.1.2

Subtract $3 x$ from $x$ .

$- 2 x + 3 = 0$

Step 3.2

Subtract $3$ from both sides of the equation.

$- 2 x = - 3$

Step 3.3

Divide each term in $- 2 x = - 3$ by $- 2$ and simplify.

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Step 3.3.1

Divide each term in $- 2 x = - 3$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 3}{- 2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 3}{- 2}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Dividing two negative values results in a positive value.

$x = \frac{3}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3}{2}$

Decimal Form:

$x = 1.5$

Mixed Number Form:

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