Precalculus Examples

$\ln (256 \sqrt{65}) (x \cdot 7)$

Step 1

Rewrite $\ln (256 \sqrt{65})$ as $\ln (256) + \ln (\sqrt{65})$ .

$(\ln (256) + \ln (\sqrt{65})) (x \cdot 7)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{65}$ as $65^{\frac{1}{2}}$ .

$(\ln (256) + \ln (65^{\frac{1}{2}})) (x \cdot 7)$

Step 3

Expand $\ln (65^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$(\ln (256) + \frac{1}{2} \ln (65)) (x \cdot 7)$

Step 4

Rewrite $\ln (65)$ as $\ln (5 (13))$ .

$(\ln (256) + \frac{1}{2} \ln (5 (13))) (x \cdot 7)$

Step 5

Rewrite $\ln (5 (13))$ as $\ln (5) + \ln (13)$ .

$(\ln (256) + \frac{1}{2} (\ln (5) + \ln (13))) (x \cdot 7)$

Step 6

Apply the distributive property.

$(\ln (256) + \frac{1}{2} \ln (5) + \frac{1}{2} \ln (13)) (x \cdot 7)$

Step 7

Combine $\frac{1}{2}$ and $\ln (5)$ .

$(\ln (256) + \frac{\ln (5)}{2} + \frac{1}{2} \ln (13)) (x \cdot 7)$

Step 8

Combine $\frac{1}{2}$ and $\ln (13)$ .

$(\ln (256) + \frac{\ln (5)}{2} + \frac{\ln (13)}{2}) (x \cdot 7)$

Step 9

Simplify each term.

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Step 9.1

Rewrite $\ln (256)$ as $\ln (2^{8})$ .

$(\ln (2^{8}) + \frac{\ln (5)}{2} + \frac{\ln (13)}{2}) (x \cdot 7)$

Step 9.2

Expand $\ln (2^{8})$ by moving $8$ outside the logarithm.

$(8 \ln (2) + \frac{\ln (5)}{2} + \frac{\ln (13)}{2}) (x \cdot 7)$

Step 10

Move $7$ to the left of $x$ .

$(8 \ln (2) + \frac{\ln (5)}{2} + \frac{\ln (13)}{2}) (7 \cdot x)$

Step 11

Apply the distributive property.

$8 \ln (2) (7 x) + \frac{\ln (5)}{2} (7 x) + \frac{\ln (13)}{2} (7 x)$

Step 12

Simplify.

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Step 12.1

Multiply $7$ by $8$ .

$56 \ln (2) x + \frac{\ln (5)}{2} (7 x) + \frac{\ln (13)}{2} (7 x)$

Step 12.2

Multiply $\frac{\ln (5)}{2} (7 x)$ .

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Step 12.2.1

Combine $7$ and $\frac{\ln (5)}{2}$ .

$56 \ln (2) x + \frac{7 \ln (5)}{2} x + \frac{\ln (13)}{2} (7 x)$

Step 12.2.2

Combine $\frac{7 \ln (5)}{2}$ and $x$ .

$56 \ln (2) x + \frac{7 \ln (5) x}{2} + \frac{\ln (13)}{2} (7 x)$

Step 12.3

Multiply $\frac{\ln (13)}{2} (7 x)$ .

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Step 12.3.1

Combine $7$ and $\frac{\ln (13)}{2}$ .

$56 \ln (2) x + \frac{7 \ln (5) x}{2} + \frac{7 \ln (13)}{2} x$

Step 12.3.2

Combine $\frac{7 \ln (13)}{2}$ and $x$ .

$56 \ln (2) x + \frac{7 \ln (5) x}{2} + \frac{7 \ln (13) x}{2}$

Step 13

Reorder factors in $56 \ln (2) x + \frac{7 \ln (5) x}{2} + \frac{7 \ln (13) x}{2}$ .

$56 x \ln (2) + \frac{7 x \ln (5)}{2} + \frac{7 x \ln (13)}{2}$