Precalculus Examples

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

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

Multiply $2$ by $2$ .

${\log_{2}}^{2} (x^{2}) - \log_{2} (x^{4})$

Step 2

Simplify each term.

Tap for more steps...

Step 2.1

Rewrite $\log_{2} (x^{2})$ using the change of base formula.

Tap for more steps...

Step 2.1.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

${\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}^{2} - \log_{2} (x^{4})$

Step 2.1.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

${(\frac{\log (x^{2})}{\log (2)})}^{2} - \log_{2} (x^{4})$

Step 2.2

Apply the product rule to $\frac{\log (x^{2})}{\log (2)}$ .

$\frac{\log^{2} (x^{2})}{\log^{2} (2)} - \log_{2} (x^{4})$

Step 2.3

Rewrite $\log_{2} (x^{4})$ using the change of base formula.

Tap for more steps...

Step 2.3.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\log^{2} (x^{2})}{\log^{2} (2)} - \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 2.3.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

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