Calculus Examples

$\ln (\sqrt{10 t^{25} v^{2}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 t^{25} v^{2}}$ as ${(10 t^{25} v^{2})}^{\frac{1}{2}}$ .

$\ln ({(10 t^{25} v^{2})}^{\frac{1}{2}})$

Step 2

Expand $\ln ({(10 t^{25} v^{2})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \ln (10 t^{25} v^{2})$

Step 3

Rewrite $\ln (10 t^{25} v^{2})$ as $\ln (10 t^{25}) + \ln (v^{2})$ .

$\frac{1}{2} (\ln (10 t^{25}) + \ln (v^{2}))$

Step 4

Rewrite $\ln (10 t^{25})$ as $\ln (10) + \ln (t^{25})$ .

$\frac{1}{2} (\ln (10) + \ln (t^{25}) + \ln (v^{2}))$

Step 5

Expand $\ln (t^{25})$ by moving $25$ outside the logarithm.

$\frac{1}{2} (\ln (10) + 25 \ln (t) + \ln (v^{2}))$

Step 6

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

$\frac{1}{2} (\ln (10) + 25 \ln (t) + 2 \ln (v))$

Step 7

Simplify each term.

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Step 7.1

Rewrite $\ln (10)$ as $\ln (2 (5))$ .

$\frac{1}{2} (\ln (2 (5)) + 25 \ln (t) + 2 \ln (v))$

Step 7.2

Rewrite $\ln (2 (5))$ as $\ln (2) + \ln (5)$ .

$\frac{1}{2} (\ln (2) + \ln (5) + 25 \ln (t) + 2 \ln (v))$

Step 8

Apply the distributive property.

$\frac{1}{2} \ln (2) + \frac{1}{2} \ln (5) + \frac{1}{2} (25 \ln (t)) + \frac{1}{2} (2 \ln (v))$

Step 9

Simplify.

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Step 9.1

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

$\frac{\ln (2)}{2} + \frac{1}{2} \ln (5) + \frac{1}{2} (25 \ln (t)) + \frac{1}{2} (2 \ln (v))$

Step 9.2

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

$\frac{\ln (2)}{2} + \frac{\ln (5)}{2} + \frac{1}{2} (25 \ln (t)) + \frac{1}{2} (2 \ln (v))$

Step 9.3

Multiply $\frac{1}{2} (25 \ln (t))$ .

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Step 9.3.1

Combine $25$ and $\frac{1}{2}$ .

$\frac{\ln (2)}{2} + \frac{\ln (5)}{2} + \frac{25}{2} \ln (t) + \frac{1}{2} (2 \ln (v))$

Step 9.3.2

Combine $\frac{25}{2}$ and $\ln (t)$ .

$\frac{\ln (2)}{2} + \frac{\ln (5)}{2} + \frac{25 \ln (t)}{2} + \frac{1}{2} (2 \ln (v))$

Step 9.4

Cancel the common factor of $2$ .

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Step 9.4.1

Factor $2$ out of $2 \ln (v)$ .

$\frac{\ln (2)}{2} + \frac{\ln (5)}{2} + \frac{25 \ln (t)}{2} + \frac{1}{2} (2 (\ln (v)))$

Step 9.4.2

Cancel the common factor.

$\frac{\ln (2)}{2} + \frac{\ln (5)}{2} + \frac{25 \ln (t)}{2} + \frac{1}{2} (2 \ln (v))$

Step 9.4.3

Rewrite the expression.

$\frac{\ln (2)}{2} + \frac{\ln (5)}{2} + \frac{25 \ln (t)}{2} + \ln (v)$