Precalculus Examples

$3 (\log (7) - \log (3) - 3 \log (x) + \frac{1}{2} \cdot \log (y))$

Step 1

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

$3 (\log (\frac{7}{3}) - 3 \log (x) + \frac{1}{2} \cdot \log (y))$

Step 2

Simplify each term.

Tap for more steps...

Step 2.1

Simplify $- 3 \log (x)$ by moving $3$ inside the logarithm.

$3 (\log (\frac{7}{3}) - \log (x^{3}) + \frac{1}{2} \cdot \log (y))$

Step 2.2

Simplify $\frac{1}{2} \log (y)$ by moving $\frac{1}{2}$ inside the logarithm.

$3 (\log (\frac{7}{3}) - \log (x^{3}) + \log (y^{\frac{1}{2}}))$

Step 3

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

$3 (\log (\frac{\frac{7}{3}}{x^{3}}) + \log (y^{\frac{1}{2}}))$

Step 4

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

$3 \log (\frac{\frac{7}{3}}{x^{3}} y^{\frac{1}{2}})$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$3 \log (\frac{7}{3} \cdot \frac{1}{x^{3}} y^{\frac{1}{2}})$

Step 6

Combine.

$3 \log (\frac{7 \cdot 1}{3 x^{3}} y^{\frac{1}{2}})$

Step 7

Multiply $7$ by $1$ .

$3 \log (\frac{7}{3 x^{3}} y^{\frac{1}{2}})$

Step 8

Combine $\frac{7}{3 x^{3}}$ and $y^{\frac{1}{2}}$ .

$3 \log (\frac{7 y^{\frac{1}{2}}}{3 x^{3}})$

Step 9

Simplify $3 \log (\frac{7 y^{\frac{1}{2}}}{3 x^{3}})$ by moving $3$ inside the logarithm.

$\log ({(\frac{7 y^{\frac{1}{2}}}{3 x^{3}})}^{3})$

Step 10

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 10.1

Apply the product rule to $\frac{7 y^{\frac{1}{2}}}{3 x^{3}}$ .

$\log (\frac{{(7 y^{\frac{1}{2}})}^{3}}{{(3 x^{3})}^{3}})$

Step 10.2

Apply the product rule to $7 y^{\frac{1}{2}}$ .

$\log (\frac{7^{3} {(y^{\frac{1}{2}})}^{3}}{{(3 x^{3})}^{3}})$

Step 10.3

Apply the product rule to $3 x^{3}$ .

$\log (\frac{7^{3} {(y^{\frac{1}{2}})}^{3}}{3^{3} {(x^{3})}^{3}})$

Step 11

Simplify the numerator.

Tap for more steps...

Step 11.1

Raise $7$ to the power of $3$ .

$\log (\frac{343 {(y^{\frac{1}{2}})}^{3}}{3^{3} {(x^{3})}^{3}})$

Step 11.2

Multiply the exponents in ${(y^{\frac{1}{2}})}^{3}$ .

Tap for more steps...

Step 11.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log (\frac{343 y^{\frac{1}{2} \cdot 3}}{3^{3} {(x^{3})}^{3}})$

Step 11.2.2

Combine $\frac{1}{2}$ and $3$ .

$\log (\frac{343 y^{\frac{3}{2}}}{3^{3} {(x^{3})}^{3}})$

Step 12

Simplify the denominator.

Tap for more steps...

Step 12.1

Raise $3$ to the power of $3$ .

$\log (\frac{343 y^{\frac{3}{2}}}{27 {(x^{3})}^{3}})$

Step 12.2

Multiply the exponents in ${(x^{3})}^{3}$ .

Tap for more steps...

Step 12.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log (\frac{343 y^{\frac{3}{2}}}{27 x^{3 \cdot 3}})$

Step 12.2.2

Multiply $3$ by $3$ .

$\log (\frac{343 y^{\frac{3}{2}}}{27 x^{9}})$