Precalculus Examples

$3 \log (x + 4) + 2 \log (x - 7) - 5 \log (x - 2) + \log (x^{2})$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log (x + 4)$ by moving $3$ inside the logarithm.

$\log ({(x + 4)}^{3}) + 2 \log (x - 7) - 5 \log (x - 2) + \log (x^{2})$

Step 1.2

Simplify $2 \log (x - 7)$ by moving $2$ inside the logarithm.

$\log ({(x + 4)}^{3}) + \log ({(x - 7)}^{2}) - 5 \log (x - 2) + \log (x^{2})$

Step 1.3

Simplify $- 5 \log (x - 2)$ by moving $5$ inside the logarithm.

$\log ({(x + 4)}^{3}) + \log ({(x - 7)}^{2}) - \log ({(x - 2)}^{5}) + \log (x^{2})$

Step 2

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

$\log ({(x + 4)}^{3} {(x - 7)}^{2}) - \log ({(x - 2)}^{5}) + \log (x^{2})$

Step 3

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

$\log (\frac{{(x + 4)}^{3} {(x - 7)}^{2}}{{(x - 2)}^{5}}) + \log (x^{2})$

Step 4

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

$\log (\frac{{(x + 4)}^{3} {(x - 7)}^{2}}{{(x - 2)}^{5}} x^{2})$

Step 5

Combine $\frac{{(x + 4)}^{3} {(x - 7)}^{2}}{{(x - 2)}^{5}}$ and $x^{2}$ .

$\log (\frac{{(x + 4)}^{3} {(x - 7)}^{2} x^{2}}{{(x - 2)}^{5}})$

Step 6

Reorder factors in $\log (\frac{{(x + 4)}^{3} {(x - 7)}^{2} x^{2}}{{(x - 2)}^{5}})$ .

$\log (\frac{x^{2} {(x + 4)}^{3} {(x - 7)}^{2}}{{(x - 2)}^{5}})$