Precalculus Examples

$2 \log_{p} (r) + \frac{1}{2} \cdot \log_{p} (z) - \frac{3}{2} \cdot \log_{p} (y) - 2 \log_{p} (a)$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log_{p} (r)$ by moving $2$ inside the logarithm.

$\log_{p} (r^{2}) + \frac{1}{2} \cdot \log_{p} (z) - \frac{3}{2} \cdot \log_{p} (y) - 2 \log_{p} (a)$

Step 1.2

Simplify $\frac{1}{2} \log_{p} (z)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{p} (r^{2}) + \log_{p} (z^{\frac{1}{2}}) - \frac{3}{2} \cdot \log_{p} (y) - 2 \log_{p} (a)$

Step 1.3

Multiply $- \frac{3}{2} \log_{p} (y)$ .

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Step 1.3.1

Reorder $\log_{p} (y)$ and $\frac{3}{2}$ .

$\log_{p} (r^{2}) + \log_{p} (z^{\frac{1}{2}}) - (\frac{3}{2} \log_{p} (y)) - 2 \log_{p} (a)$

Step 1.3.2

Simplify $\frac{3}{2} \log_{p} (y)$ by moving $\frac{3}{2}$ inside the logarithm.

$\log_{p} (r^{2}) + \log_{p} (z^{\frac{1}{2}}) - \log_{p} (y^{\frac{3}{2}}) - 2 \log_{p} (a)$

Step 1.4

Simplify $- 2 \log_{p} (a)$ by moving $2$ inside the logarithm.

$\log_{p} (r^{2}) + \log_{p} (z^{\frac{1}{2}}) - \log_{p} (y^{\frac{3}{2}}) - \log_{p} (a^{2})$

Step 2

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

$\log_{p} (r^{2} z^{\frac{1}{2}}) - \log_{p} (y^{\frac{3}{2}}) - \log_{p} (a^{2})$

Step 3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{p} (\frac{r^{2} z^{\frac{1}{2}}}{y^{\frac{3}{2}}}) - \log_{p} (a^{2})$

Step 4

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{p} (\frac{\frac{r^{2} z^{\frac{1}{2}}}{y^{\frac{3}{2}}}}{a^{2}})$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\log_{p} (\frac{r^{2} z^{\frac{1}{2}}}{y^{\frac{3}{2}}} \cdot \frac{1}{a^{2}})$

Step 6

Combine.

$\log_{p} (\frac{r^{2} z^{\frac{1}{2}} \cdot 1}{y^{\frac{3}{2}} a^{2}})$

Step 7

Simplify the expression.

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Step 7.1

Multiply $r^{2}$ by $1$ .

$\log_{p} (\frac{r^{2} z^{\frac{1}{2}}}{y^{\frac{3}{2}} a^{2}})$

Step 7.2

Reorder factors in $\log_{p} (\frac{r^{2} z^{\frac{1}{2}}}{y^{\frac{3}{2}} a^{2}})$ .

$\log_{p} (\frac{r^{2} z^{\frac{1}{2}}}{a^{2} y^{\frac{3}{2}}})$