Precalculus Examples

$\log (x^{2} \cdot 49) - 3 \log (x + 7) + 5 \log (x)$

Step 1

Simplify each term.

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Step 1.1

Move $49$ to the left of $x^{2}$ .

$\log (49 \cdot x^{2}) - 3 \log (x + 7) + 5 \log (x)$

Step 1.2

Simplify $- 3 \log (x + 7)$ by moving $3$ inside the logarithm.

$\log (49 x^{2}) - \log ({(x + 7)}^{3}) + 5 \log (x)$

Step 1.3

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

$\log (49 x^{2}) - \log ({(x + 7)}^{3}) + \log (x^{5})$

Step 2

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

$\log (\frac{49 x^{2}}{{(x + 7)}^{3}}) + \log (x^{5})$

Step 3

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

$\log (\frac{49 x^{2}}{{(x + 7)}^{3}} x^{5})$

Step 4

Multiply $\frac{49 x^{2}}{{(x + 7)}^{3}} x^{5}$ .

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Step 4.1

Combine $\frac{49 x^{2}}{{(x + 7)}^{3}}$ and $x^{5}$ .

$\log (\frac{49 x^{2} x^{5}}{{(x + 7)}^{3}})$

Step 4.2

Multiply $x^{2}$ by $x^{5}$ by adding the exponents.

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Step 4.2.1

Move $x^{5}$ .

$\log (\frac{49 (x^{5} x^{2})}{{(x + 7)}^{3}})$

Step 4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log (\frac{49 x^{5 + 2}}{{(x + 7)}^{3}})$

Step 4.2.3

Add $5$ and $2$ .

$\log (\frac{49 x^{7}}{{(x + 7)}^{3}})$