Precalculus Examples

$\log (10) (x^{2} - 81) - 4 \log (10) (x + 9) + 5 \log (10 (x))$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Logarithm base $10$ of $10$ is $1$ .

$1 (x^{2} - 81) - 4 \log (10) (x + 9) + 5 \log (10 (x))$

Step 1.2

Multiply $x^{2} - 81$ by $1$ .

$x^{2} - 81 - 4 \log (10) (x + 9) + 5 \log (10 (x))$

Step 1.3

Logarithm base $10$ of $10$ is $1$ .

$x^{2} - 81 - 4 \cdot 1 (x + 9) + 5 \log (10 (x))$

Step 1.4

Multiply $- 4$ by $1$ .

$x^{2} - 81 - 4 (x + 9) + 5 \log (10 (x))$

Step 1.5

Apply the distributive property.

$x^{2} - 81 - 4 x - 4 \cdot 9 + 5 \log (10 (x))$

Step 1.6

Multiply $- 4$ by $9$ .

$x^{2} - 81 - 4 x - 36 + 5 \log (10 (x))$

Step 1.7

Simplify $5 \log (10 x)$ by moving $5$ inside the logarithm.

$x^{2} - 81 - 4 x - 36 + \log ({(10 x)}^{5})$

Step 1.8

Apply the product rule to $10 x$ .

$x^{2} - 81 - 4 x - 36 + \log (10^{5} x^{5})$

Step 1.9

Rewrite $\log (10^{5} x^{5})$ as $\log (10^{5}) + \log (x^{5})$ .

$x^{2} - 81 - 4 x - 36 + \log (10^{5}) + \log (x^{5})$

Step 1.10

Use logarithm rules to move $5$ out of the exponent.

$x^{2} - 81 - 4 x - 36 + 5 \log (10) + \log (x^{5})$

Step 1.11

Logarithm base $10$ of $10$ is $1$ .

$x^{2} - 81 - 4 x - 36 + 5 \cdot 1 + \log (x^{5})$

Step 1.12

Multiply $5$ by $1$ .

$x^{2} - 81 - 4 x - 36 + 5 + \log (x^{5})$

Step 2

Simplify by adding and subtracting.

Tap for more steps...

Step 2.1

Subtract $36$ from $- 81$ .

$x^{2} - 4 x - 117 + 5 + \log (x^{5})$

Step 2.2

Add $- 117$ and $5$ .

$x^{2} - 4 x - 112 + \log (x^{5})$