Precalculus Examples

$2 \log_{5} (5 x) - \log_{5} ({(x)}^{2})$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Simplify $2 \log_{5} (5 x)$ by moving $2$ inside the logarithm.

$\log_{5} ({(5 x)}^{2}) - \log_{5} ({(x)}^{2})$

Step 1.2

Apply the product rule to $5 x$ .

$\log_{5} (5^{2} x^{2}) - \log_{5} ({(x)}^{2})$

Step 1.3

Rewrite $\log_{5} (5^{2} x^{2})$ as $\log_{5} (5^{2}) + \log_{5} (x^{2})$ .

$\log_{5} (5^{2}) + \log_{5} (x^{2}) - \log_{5} ({(x)}^{2})$

Step 1.4

Use logarithm rules to move $2$ out of the exponent.

$2 \log_{5} (5) + \log_{5} (x^{2}) - \log_{5} ({(x)}^{2})$

Step 1.5

Logarithm base $5$ of $5$ is $1$ .

$2 \cdot 1 + \log_{5} (x^{2}) - \log_{5} ({(x)}^{2})$

Step 1.6

Multiply $2$ by $1$ .

$2 + \log_{5} (x^{2}) - \log_{5} (x^{2})$

Step 2

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

$2 + \log_{5} (\frac{x^{2}}{x^{2}})$

Step 3

Simplify each term.

Tap for more steps...

Step 3.1

Cancel the common factor of $x^{2}$ .

Tap for more steps...

Step 3.1.1

Cancel the common factor.

$2 + \log_{5} (\frac{x^{2}}{x^{2}})$

Step 3.1.2

Rewrite the expression.

$2 + \log_{5} (1)$

Step 3.2

Logarithm base $5$ of $1$ is $0$ .

$2 + 0$

Step 4

Add $2$ and $0$ .

$2$