Calculus Examples

$3 \log (x) - \frac{1}{2} \log (z) - \log (y + 1) + \log (12)$

Step 1

Simplify each term.

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Step 1.1

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

$\log (x^{3}) - \frac{1}{2} \cdot \log (z) - \log (y + 1) + \log (12)$

Step 1.2

Multiply $- \frac{1}{2} \log (z)$ .

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Step 1.2.1

Reorder $\log (z)$ and $\frac{1}{2}$ .

$\log (x^{3}) - (\frac{1}{2} \log (z)) - \log (y + 1) + \log (12)$

Step 1.2.2

Simplify $\frac{1}{2} \log (z)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log (x^{3}) - \log (z^{\frac{1}{2}}) - \log (y + 1) + \log (12)$

Step 2

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

$\log (\frac{x^{3}}{z^{\frac{1}{2}}}) - \log (y + 1) + \log (12)$

Step 3

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

$\log (\frac{\frac{x^{3}}{z^{\frac{1}{2}}}}{y + 1}) + \log (12)$

Step 4

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

$\log (\frac{\frac{x^{3}}{z^{\frac{1}{2}}}}{y + 1} \cdot 12)$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{x^{3}}{z^{\frac{1}{2}}} \cdot \frac{1}{y + 1} \cdot 12)$

Step 6

Multiply $\frac{x^{3}}{z^{\frac{1}{2}}}$ by $\frac{1}{y + 1}$ .

$\log (\frac{x^{3}}{z^{\frac{1}{2}} (y + 1)} \cdot 12)$

Step 7

Combine $\frac{x^{3}}{z^{\frac{1}{2}} (y + 1)}$ and $12$ .

$\log (\frac{x^{3} \cdot 12}{z^{\frac{1}{2}} (y + 1)})$

Step 8

Move $12$ to the left of $x^{3}$ .

$\log (\frac{12 x^{3}}{z^{\frac{1}{2}} (y + 1)})$