Precalculus Examples

$\log (3) (2 x + 1) + \log (3) (x + 4) = 2$

Step 1

Simplify the left side.

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

Apply the distributive property.

$\log (3) (2 x) + \log (3) \cdot 1 + \log (3) (x + 4) = 2$

Step 1.1.2

Multiply $\log (3)$ by $1$ .

$\log (3) \cdot (2 x) + \log (3) + \log (3) (x + 4) = 2$

Step 1.1.3

Move $2$ to the left of $\log (3)$ .

$2 \log (3) x + \log (3) + \log (3) (x + 4) = 2$

Step 1.1.4

Apply the distributive property.

$2 \log (3) x + \log (3) + \log (3) x + \log (3) \cdot 4 = 2$

Step 1.1.5

Move $4$ to the left of $\log (3)$ .

$2 \log (3) x + \log (3) + \log (3) x + 4 \log (3) = 2$

Step 1.2

Simplify by adding terms.

Tap for more steps...

Step 1.2.1

Add $2 \log (3) x$ and $\log (3) x$ .

$3 \log (3) x + \log (3) + 4 \log (3) = 2$

Step 1.2.2

Add $\log (3)$ and $4 \log (3)$ .

$3 \log (3) x + 5 \log (3) = 2$

Step 1.2.3

Reorder factors in $3 \log (3) x + 5 \log (3)$ .

$3 x \log (3) + 5 \log (3) = 2$

Step 2

Simplify the left side.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

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

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

Step 2.1.2

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

$x \log (27) + 5 \log (3) = 2$

Step 2.1.3

Simplify $5 \log (3)$ by moving $5$ inside the logarithm.

$x \log (27) + \log (3^{5}) = 2$

Step 2.1.4

Raise $3$ to the power of $5$ .

$x \log (27) + \log (243) = 2$

Step 3

Subtract $\log (243)$ from both sides of the equation.

$x \log (27) = 2 - \log (243)$

Step 4

Divide each term in $x \log (27) = 2 - \log (243)$ by $\log (27)$ and simplify.

Tap for more steps...

Step 4.1

Divide each term in $x \log (27) = 2 - \log (243)$ by $\log (27)$ .

$\frac{x \log (27)}{\log (27)} = \frac{2}{\log (27)} + \frac{- \log (243)}{\log (27)}$

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Cancel the common factor of $\log (27)$ .

Tap for more steps...

Step 4.2.1.1

Cancel the common factor.

$\frac{x \log (27)}{\log (27)} = \frac{2}{\log (27)} + \frac{- \log (243)}{\log (27)}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{\log (27)} + \frac{- \log (243)}{\log (27)}$

Step 4.3

Simplify the right side.

Tap for more steps...

Step 4.3.1

Move the negative in front of the fraction.

$x = \frac{2}{\log (27)} - \frac{\log (243)}{\log (27)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2}{\log (27)} - \frac{\log (243)}{\log (27)}$

Decimal Form:

$x = - 0.26939781 \dots$