Precalculus Examples

$2 \log (2 x) - \log (y) - \log (3 + 2) \log (5)$

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Apply the product rule to $2 x$ .

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

Step 1.3

Raise $2$ to the power of $2$ .

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

Step 1.4

Add $3$ and $2$ .

$\log (4 x^{2}) - \log (y) - \log (5) \log (5)$

Step 1.5

Multiply $- \log (5) \log (5)$ .

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Step 1.5.1

Raise $\log (5)$ to the power of $1$ .

$\log (4 x^{2}) - \log (y) - (\log^{1} (5) \log (5))$

Step 1.5.2

Raise $\log (5)$ to the power of $1$ .

$\log (4 x^{2}) - \log (y) - (\log^{1} (5) \log^{1} (5))$

Step 1.5.3

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

$\log (4 x^{2}) - \log (y) - {\log (5)}^{1 + 1}$

Step 1.5.4

Add $1$ and $1$ .

$\log (4 x^{2}) - \log (y) - \log^{2} (5)$

Step 2

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

$\log (\frac{4 x^{2}}{y}) - \log^{2} (5)$