Precalculus Examples

$\log_{3} ({(\frac{4 x}{7 y})}^{9})$

Step 1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1

Apply the product rule to $\frac{4 x}{7 y}$ .

$\log_{3} (\frac{{(4 x)}^{9}}{{(7 y)}^{9}})$

Step 1.2

Apply the product rule to $4 x$ .

$\log_{3} (\frac{4^{9} x^{9}}{{(7 y)}^{9}})$

Step 1.3

Apply the product rule to $7 y$ .

$\log_{3} (\frac{4^{9} x^{9}}{7^{9} y^{9}})$

Step 2

Raise $4$ to the power of $9$ .

$\log_{3} (\frac{262144 x^{9}}{7^{9} y^{9}})$

Step 3

Raise $7$ to the power of $9$ .

$\log_{3} (\frac{262144 x^{9}}{40353607 y^{9}})$

Step 4

Rewrite $\log_{3} (\frac{262144 x^{9}}{40353607 y^{9}})$ using the change of base formula.

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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 $0$ .

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

Step 4.2

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

$\frac{\log (\frac{262144 x^{9}}{40353607 y^{9}})}{\log (3)}$