Precalculus Examples

$\ln (\frac{y^{2}}{z \sqrt{x^{2} + 25}})$

Step 1

Rewrite $\ln (\frac{y^{2}}{z \sqrt{x^{2} + 25}})$ as $\ln (y^{2}) - \ln (z \sqrt{x^{2} + 25})$ .

$\ln (y^{2}) - \ln (z \sqrt{x^{2} + 25})$

Step 2

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

$\ln (y^{2}) - \ln (z {(x^{2} + 25)}^{\frac{1}{2}})$

Step 3

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

$2 \ln (y) - \ln (z {(x^{2} + 25)}^{\frac{1}{2}})$

Step 4

Rewrite $\ln (z {(x^{2} + 25)}^{\frac{1}{2}})$ as $\ln (z) + \ln ({(x^{2} + 25)}^{\frac{1}{2}})$ .

$2 \ln (y) - (\ln (z) + \ln ({(x^{2} + 25)}^{\frac{1}{2}}))$

Step 5

Expand $\ln ({(x^{2} + 25)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$2 \ln (y) - (\ln (z) + \frac{1}{2} \ln (x^{2} + 25))$

Step 6

Simplify each term.

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Step 6.1

Combine $\frac{1}{2}$ and $\ln (x^{2} + 25)$ .

$2 \ln (y) - (\ln (z) + \frac{\ln (x^{2} + 25)}{2})$

Step 6.2

To write $\ln (z)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 \ln (y) - (\ln (z) \cdot \frac{2}{2} + \frac{\ln (x^{2} + 25)}{2})$

Step 6.3

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

$2 \ln (y) - (\frac{\ln (z) \cdot 2}{2} + \frac{\ln (x^{2} + 25)}{2})$

Step 6.4

Combine the numerators over the common denominator.

$2 \ln (y) - \frac{\ln (z) \cdot 2 + \ln (x^{2} + 25)}{2}$

Step 6.5

Move $2$ to the left of $\ln (z)$ .

$2 \ln (y) - \frac{2 \ln (z) + \ln (x^{2} + 25)}{2}$