Precalculus Examples

$2 \log_{4} (x - (3 \log_{4} (y) + \log_{4} (z)))$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log_{4} (y)$ by moving $3$ inside the logarithm.

$2 \log_{4} (x - (\log_{4} (y^{3}) + \log_{4} (z)))$

Step 1.2

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

$2 \log_{4} (x - \log_{4} (y^{3} z))$

Step 2

Simplify $2 \log_{4} (x - \log_{4} (y^{3} z))$ by moving $2$ inside the logarithm.

$\log_{4} ({(x - \log_{4} (y^{3} z))}^{2})$

Step 3

Rewrite $\log_{4} ({(x - \log_{4} (y^{3} z))}^{2})$ using the change of base formula.

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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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 3.2

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

$\frac{\log ({(x - \log_{4} (y^{3} z))}^{2})}{\log (4)}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $\log_{4} (y^{3} z)$ using the change of base formula.

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Step 4.1.1

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 $0$ .

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

Step 4.1.2

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

$\frac{\log ({(x - \frac{\log (y^{3} z)}{\log (4)})}^{2})}{\log (4)}$

Step 4.2

To write $x$ as a fraction with a common denominator, multiply by $\frac{\log (4)}{\log (4)}$ .

$\frac{\log ({(\frac{x \log (4)}{\log (4)} - \frac{\log (y^{3} z)}{\log (4)})}^{2})}{\log (4)}$

Step 4.3

Combine the numerators over the common denominator.

$\frac{\log ({(\frac{x \log (4) - \log (y^{3} z)}{\log (4)})}^{2})}{\log (4)}$

Step 4.4

Apply the product rule to $\frac{x \log (4) - \log (y^{3} z)}{\log (4)}$ .

$\frac{\log (\frac{{(x \log (4) - \log (y^{3} z))}^{2}}{\log^{2} (4)})}{\log (4)}$