Precalculus Examples

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

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

Step 1

Simplify each term.

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Step 1.1

Simplify

3 \log_{4} (y)

$3 \log_{4} (y)$ by moving

3

$3$ inside the logarithm.

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

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

Step 1.2

Use the product property of logarithms,

\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)

$\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

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

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

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

Step 2

Simplify

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

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

2

$2$ inside the logarithm.

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

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

Step 3

Rewrite

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

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

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Step 3.1

The change of base rule can be used if

a

$a$ and

b

$b$ are greater than

0

$0$ and not equal to

1

$1$ , and

x

$x$ is greater than

0

$0$ .

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

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

Step 3.2

Substitute in values for the variables in the change of base formula, using

b = 10

$b = 10$ .

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

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

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

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

Step 4

Simplify the numerator.

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Step 4.1

Rewrite

\log_{4} (y^{3} z)

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

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Step 4.1.1

The change of base rule can be used if

a

$a$ and

b

$b$ are greater than

0

$0$ and not equal to

1

$1$ , and

x

$x$ is greater than

0

$0$ .

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

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

Step 4.1.2

Substitute in values for the variables in the change of base formula, using

b = 10

$b = 10$ .

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

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

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

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

Step 4.2

To write

x

$x$ as a fraction with a common denominator, multiply by

\frac{\log (4)}{\log (4)}

$\frac{\log (4)}{\log (4)}$ .

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

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

Step 4.4

Apply the product rule to

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

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

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

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

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

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