Precalculus Examples

$\log_{4} ({(3)}^{4}) + \log_{4} (9)$

Step 1

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

$\log_{4} ({(3)}^{4} \cdot 9)$

Step 2

Raise $3$ to the power of $4$ .

$\log_{4} (81 \cdot 9)$

Step 3

Multiply $81$ by $9$ .

$\log_{4} (729)$

Step 4

Rewrite $\log_{4} (729)$ 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 $0$ .

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

Step 4.2

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

$\frac{\log (729)}{\log (4)}$

Step 5

Rewrite $\log (729)$ as $\log (3^{6})$ .

$\frac{\log (3^{6})}{\log (4)}$

Step 6

Expand $\log (3^{6})$ by moving $6$ outside the logarithm.

$\frac{6 \log (3)}{\log (4)}$

Step 7

Rewrite $\log (4)$ as $\log (2^{2})$ .

$\frac{6 \log (3)}{\log (2^{2})}$

Step 8

Expand $\log (2^{2})$ by moving $2$ outside the logarithm.

$\frac{6 \log (3)}{2 \log (2)}$

Step 9

Cancel the common factor of $6$ and $2$ .

Tap for more steps...

Step 9.1

Factor $2$ out of $6 \log (3)$ .

$\frac{2 (3 \log (3))}{2 \log (2)}$

Step 9.2

Cancel the common factors.

Tap for more steps...

Step 9.2.1

Factor $2$ out of $2 \log (2)$ .

$\frac{2 (3 \log (3))}{2 (\log (2))}$

Step 9.2.2

Cancel the common factor.

$\frac{2 (3 \log (3))}{2 \log (2)}$

Step 9.2.3

Rewrite the expression.

$\frac{3 \log (3)}{\log (2)}$

Step 10

Simplify $3 \log (3)$ by moving $3$ inside the logarithm.

$\frac{\log (3^{3})}{\log (2)}$

Step 11

Raise $3$ to the power of $3$ .

$\frac{\log (27)}{\log (2)}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (27)}{\log (2)}$

Decimal Form:

$4.75488750 \dots$