Algebra Examples

3 {log}_{2} (x) - 4 {log}_{2} (x + 3) + {log}_{2} (y)

$3 \log_{2} (x) - 4 \log_{2} (x + 3) + \log_{2} (y)$

Step 1

Simplify each term.

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Step 1.1

Simplify

3 {log}_{2} (x)

$3 \log_{2} (x)$ by moving

3

$3$ inside the logarithm.

{log}_{2} (x^{3}) - 4 {log}_{2} (x + 3) + {log}_{2} (y)

$\log_{2} (x^{3}) - 4 \log_{2} (x + 3) + \log_{2} (y)$

Step 1.2

Simplify

- 4 {log}_{2} (x + 3)

$- 4 \log_{2} (x + 3)$ by moving

4

$4$ inside the logarithm.

{log}_{2} (x^{3}) - {log}_{2} ({(x + 3)}^{4}) + {log}_{2} (y)

$\log_{2} (x^{3}) - \log_{2} ({(x + 3)}^{4}) + \log_{2} (y)$

{log}_{2} (x^{3}) - {log}_{2} ({(x + 3)}^{4}) + {log}_{2} (y)

$\log_{2} (x^{3}) - \log_{2} ({(x + 3)}^{4}) + \log_{2} (y)$

Step 2

Use the quotient property of logarithms,

{log}_{b} (x) - {log}_{b} (y) = {log}_{b} (\frac{x}{y})

$\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

{log}_{2} (\frac{x^{3}}{{(x + 3)}^{4}}) + {log}_{2} (y)

$\log_{2} (\frac{x^{3}}{{(x + 3)}^{4}}) + \log_{2} (y)$

Step 3

Use the product property of logarithms,

{log}_{b} (x) + {log}_{b} (y) = {log}_{b} (x y)

$\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{2} (\frac{x^{3}}{{(x + 3)}^{4}} y)$

Step 4

Combine

$\frac{x^{3}}{{(x + 3)}^{4}}$ and

$y$ .

$\log_{2} (\frac{x^{3} y}{{(x + 3)}^{4}})$

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