Calculus Examples

$\log_{b} (x + 3) + \log_{b} (x) = \log_{b} (54)$

Step 1

Simplify the left side.

Tap for more steps...

Step 1.1

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

$\log_{b} ((x + 3) x) = \log_{b} (54)$

Step 1.2

Apply the distributive property.

$\log_{b} (x \cdot x + 3 x) = \log_{b} (54)$

Step 1.3

Multiply $x$ by $x$ .

$\log_{b} (x^{2} + 3 x) = \log_{b} (54)$

Step 2

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

$x^{2} + 3 x = 54$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Subtract $54$ from both sides of the equation.

$x^{2} + 3 x - 54 = 0$

Step 3.2

Factor $x^{2} + 3 x - 54$ using the AC method.

Tap for more steps...

Step 3.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 54$ and whose sum is $3$ .

$- 6, 9$

Step 3.2.2

Write the factored form using these integers.

$(x - 6) (x + 9) = 0$

Step 3.3

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

$x - 6 = 0$

$x + 9 = 0$

Step 3.4

Set $x - 6$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.4.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 3.4.2

Add $6$ to both sides of the equation.

$x = 6$

Step 3.5

Set $x + 9$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.5.1

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Step 3.5.2

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 3.6

The final solution is all the values that make $(x - 6) (x + 9) = 0$ true.

$x = 6, - 9$