Precalculus Examples

$\frac{1}{2} \cdot (\log_{c} (x + 2) \log_{c} (y - 4) \log_{c} (x))$

Step 1

Multiply $\frac{1}{2} (\log_{c} (x + 2) \log_{c} (y - 4) \log_{c} (x))$ .

Tap for more steps...

Step 1.1

Reorder $\log_{c} (x + 2)$ and $\frac{1}{2}$ .

$\frac{1}{2} \log_{c} (x + 2) (\log_{c} (y - 4) \log_{c} (x))$

Step 1.2

Simplify $\frac{1}{2} \log_{c} (x + 2)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{c} ({(x + 2)}^{\frac{1}{2}}) (\log_{c} (y - 4) \log_{c} (x))$

Step 2

Reorder factors in $\log_{c} ({(x + 2)}^{\frac{1}{2}}) (\log_{c} (y - 4) \log_{c} (x))$ .

$\log_{c} ({(x + 2)}^{\frac{1}{2}}) \log_{c} (y - 4) \log_{c} (x)$

Step 3

Rewrite $\log_{c} ({(x + 2)}^{\frac{1}{2}})$ using the change of base formula.

Tap for more steps...

Step 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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{c} (y - 4) \log_{c} (x)$

Step 3.2

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

$\frac{\log ({(x + 2)}^{\frac{1}{2}})}{\log (c)} \log_{c} (y - 4) \log_{c} (x)$

Step 4

Rewrite $\log_{c} (y - 4)$ using the change of base formula.

Tap for more steps...

Step 4.1

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 $0$ .

$\frac{\log ({(x + 2)}^{\frac{1}{2}})}{\log (c)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{c} (x)$

Step 4.2

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

$\frac{\log ({(x + 2)}^{\frac{1}{2}})}{\log (c)} \cdot \frac{\log (y - 4)}{\log (c)} \log_{c} (x)$

Step 5

Combine.

$\frac{\log ({(x + 2)}^{\frac{1}{2}}) \log (y - 4)}{\log (c) \log (c)} \log_{c} (x)$

Step 6

Simplify the denominator.

Tap for more steps...

Step 6.1

Raise $\log (c)$ to the power of $1$ .

$\frac{\log ({(x + 2)}^{\frac{1}{2}}) \log (y - 4)}{\log^{1} (c) \log (c)} \log_{c} (x)$

Step 6.2

Raise $\log (c)$ to the power of $1$ .

$\frac{\log ({(x + 2)}^{\frac{1}{2}}) \log (y - 4)}{\log^{1} (c) \log^{1} (c)} \log_{c} (x)$

Step 6.3

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

$\frac{\log ({(x + 2)}^{\frac{1}{2}}) \log (y - 4)}{{\log (c)}^{1 + 1}} \log_{c} (x)$

Step 6.4

Add $1$ and $1$ .

$\frac{\log ({(x + 2)}^{\frac{1}{2}}) \log (y - 4)}{\log^{2} (c)} \log_{c} (x)$

Step 7

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

Tap for more steps...

Step 7.1

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 $0$ .

$\frac{\log ({(x + 2)}^{\frac{1}{2}}) \log (y - 4)}{\log^{2} (c)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 7.2

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

$\frac{\log ({(x + 2)}^{\frac{1}{2}}) \log (y - 4)}{\log^{2} (c)} \cdot \frac{\log (x)}{\log (c)}$

Step 8

Combine.

$\frac{\log ({(x + 2)}^{\frac{1}{2}}) \log (y - 4) \log (x)}{\log^{2} (c) \log (c)}$

Step 9

Multiply $\log^{2} (c)$ by $\log (c)$ by adding the exponents.

Tap for more steps...

Step 9.1

Multiply $\log^{2} (c)$ by $\log (c)$ .

Tap for more steps...

Step 9.1.1

Raise $\log (c)$ to the power of $1$ .

$\frac{\log ({(x + 2)}^{\frac{1}{2}}) \log (y - 4) \log (x)}{\log^{2} (c) \log^{1} (c)}$

Step 9.1.2

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

$\frac{\log ({(x + 2)}^{\frac{1}{2}}) \log (y - 4) \log (x)}{{\log (c)}^{2 + 1}}$

Step 9.2

Add $2$ and $1$ .

$\frac{\log ({(x + 2)}^{\frac{1}{2}}) \log (y - 4) \log (x)}{\log^{3} (c)}$