Precalculus Examples

$2 \ln (3 x) + \ln (5 x) - \ln (6 x y)$

Step 1

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

$2 \ln (3 x) + \ln (\frac{5 x}{6 x y})$

Step 2

Simplify each term.

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Step 2.1

Simplify $2 \ln (3 x)$ by moving $2$ inside the logarithm.

$\ln ({(3 x)}^{2}) + \ln (\frac{5 x}{6 x y})$

Step 2.2

Apply the product rule to $3 x$ .

$\ln (3^{2} x^{2}) + \ln (\frac{5 x}{6 x y})$

Step 2.3

Raise $3$ to the power of $2$ .

$\ln (9 x^{2}) + \ln (\frac{5 x}{6 x y})$

Step 2.4

Cancel the common factor of $x$ .

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Step 2.4.1

Cancel the common factor.

$\ln (9 x^{2}) + \ln (\frac{5 x}{6 x y})$

Step 2.4.2

Rewrite the expression.

$\ln (9 x^{2}) + \ln (\frac{5}{6 y})$

Step 3

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

$\ln (9 x^{2} \frac{5}{6 y})$

Step 4

Cancel the common factor of $3$ .

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Step 4.1

Factor $3$ out of $9 x^{2}$ .

$\ln (3 (3 x^{2}) \frac{5}{6 y})$

Step 4.2

Factor $3$ out of $6 y$ .

$\ln (3 (3 x^{2}) \frac{5}{3 (2 y)})$

Step 4.3

Cancel the common factor.

$\ln (3 (3 x^{2}) \frac{5}{3 (2 y)})$

Step 4.4

Rewrite the expression.

$\ln (3 x^{2} \frac{5}{2 y})$

Step 5

Combine $3$ and $\frac{5}{2 y}$ .

$\ln (x^{2} \frac{3 \cdot 5}{2 y})$

Step 6

Multiply $3$ by $5$ .

$\ln (x^{2} \frac{15}{2 y})$

Step 7

Combine $x^{2}$ and $\frac{15}{2 y}$ .

$\ln (\frac{x^{2} \cdot 15}{2 y})$

Step 8

Move $15$ to the left of $x^{2}$ .

$\ln (\frac{15 x^{2}}{2 y})$