Precalculus Examples

$\ln (\frac{11 x^{2}}{9 \sqrt{y} t^{5}})$

Step 1

Rewrite $\ln (\frac{11 x^{2}}{9 \sqrt{y} t^{5}})$ as $\ln (11 x^{2}) - \ln (9 \sqrt{y} t^{5})$ .

$\ln (11 x^{2}) - \ln (9 \sqrt{y} t^{5})$

Step 2

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

$\ln (11 x^{2}) - \ln (9 y^{\frac{1}{2}} t^{5})$

Step 3

Rewrite $\ln (11 x^{2})$ as $\ln (11) + \ln (x^{2})$ .

$\ln (11) + \ln (x^{2}) - \ln (9 y^{\frac{1}{2}} t^{5})$

Step 4

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

$\ln (11) + 2 \ln (x) - \ln (9 y^{\frac{1}{2}} t^{5})$

Step 5

Rewrite $\ln (9 y^{\frac{1}{2}} t^{5})$ as $\ln (9 y^{\frac{1}{2}}) + \ln (t^{5})$ .

$\ln (11) + 2 \ln (x) - (\ln (9 y^{\frac{1}{2}}) + \ln (t^{5}))$

Step 6

Rewrite $\ln (9 y^{\frac{1}{2}})$ as $\ln (9) + \ln (y^{\frac{1}{2}})$ .

$\ln (11) + 2 \ln (x) - (\ln (9) + \ln (y^{\frac{1}{2}}) + \ln (t^{5}))$

Step 7

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

$\ln (11) + 2 \ln (x) - (\ln (9) + \frac{1}{2} \ln (y) + \ln (t^{5}))$

Step 8

Expand $\ln (t^{5})$ by moving $5$ outside the logarithm.

$\ln (11) + 2 \ln (x) - (\ln (9) + \frac{1}{2} \ln (y) + 5 \ln (t))$

Step 9

Simplify each term.

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Step 9.1

Simplify each term.

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Step 9.1.1

Rewrite $\ln (9)$ as $\ln (3^{2})$ .

$\ln (11) + 2 \ln (x) - (\ln (3^{2}) + \frac{1}{2} \ln (y) + 5 \ln (t))$

Step 9.1.2

Expand $\ln (3^{2})$ by moving $2$ outside the logarithm.

$\ln (11) + 2 \ln (x) - (2 \ln (3) + \frac{1}{2} \ln (y) + 5 \ln (t))$

Step 9.1.3

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

$\ln (11) + 2 \ln (x) - (2 \ln (3) + \frac{\ln (y)}{2} + 5 \ln (t))$

Step 9.2

Apply the distributive property.

$\ln (11) + 2 \ln (x) - (2 \ln (3)) - \frac{\ln (y)}{2} - (5 \ln (t))$

Step 9.3

Simplify.

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Step 9.3.1

Multiply $2$ by $- 1$ .

$\ln (11) + 2 \ln (x) - 2 \ln (3) - \frac{\ln (y)}{2} - (5 \ln (t))$

Step 9.3.2

Multiply $5$ by $- 1$ .

$\ln (11) + 2 \ln (x) - 2 \ln (3) - \frac{\ln (y)}{2} - 5 \ln (t)$