Precalculus Examples

$\log (x) + 4 \log ({(x)}^{2}) - 3 \log ({(x)}^{3})$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Simplify $4 \log (x^{2})$ by moving $4$ inside the logarithm.

$\log (x) + \log ({(x^{2})}^{4}) - 3 \log ({(x)}^{3})$

Step 1.2

Multiply the exponents in ${(x^{2})}^{4}$ .

Tap for more steps...

Step 1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log (x) + \log (x^{2 \cdot 4}) - 3 \log ({(x)}^{3})$

Step 1.2.2

Multiply $2$ by $4$ .

$\log (x) + \log (x^{8}) - 3 \log ({(x)}^{3})$

Step 1.3

Simplify $- 3 \log (x^{3})$ by moving $3$ inside the logarithm.

$\log (x) + \log (x^{8}) - \log ({(x^{3})}^{3})$

Step 1.4

Multiply the exponents in ${(x^{3})}^{3}$ .

Tap for more steps...

Step 1.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log (x) + \log (x^{8}) - \log (x^{3 \cdot 3})$

Step 1.4.2

Multiply $3$ by $3$ .

$\log (x) + \log (x^{8}) - \log (x^{9})$

Step 2

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

$\log (x \cdot x^{8}) - \log (x^{9})$

Step 3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{x \cdot x^{8}}{x^{9}})$

Step 4

Multiply $x$ by $x^{8}$ by adding the exponents.

Tap for more steps...

Step 4.1

Multiply $x$ by $x^{8}$ .

Tap for more steps...

Step 4.1.1

Raise $x$ to the power of $1$ .

$\log (\frac{x^{1} x^{8}}{x^{9}})$

Step 4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log (\frac{x^{1 + 8}}{x^{9}})$

Step 4.2

Add $1$ and $8$ .

$\log (\frac{x^{9}}{x^{9}})$

Step 5

Cancel the common factor of $x^{9}$ .

Tap for more steps...

Step 5.1

Cancel the common factor.

$\log (\frac{x^{9}}{x^{9}})$

Step 5.2

Rewrite the expression.

$\log (1)$

Step 6

Logarithm base $10$ of $1$ is $0$ .

$0$