Precalculus Examples

$2 \ln (4) - 4 \ln (x) + 0.5 \ln (y) - 4 \ln (z) + 6 \ln (w) + 3 \ln (t)$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \ln (4)$ by moving $2$ inside the logarithm.

$\ln (4^{2}) - 4 \ln (x) + 0.5 \ln (y) - 4 \ln (z) + 6 \ln (w) + 3 \ln (t)$

Step 1.2

Raise $4$ to the power of $2$ .

$\ln (16) - 4 \ln (x) + 0.5 \ln (y) - 4 \ln (z) + 6 \ln (w) + 3 \ln (t)$

Step 1.3

Simplify $- 4 \ln (x)$ by moving $4$ inside the logarithm.

$\ln (16) - \ln (x^{4}) + 0.5 \ln (y) - 4 \ln (z) + 6 \ln (w) + 3 \ln (t)$

Step 1.4

Simplify $0.5 \ln (y)$ by moving $0.5$ inside the logarithm.

$\ln (16) - \ln (x^{4}) + \ln (y^{0.5}) - 4 \ln (z) + 6 \ln (w) + 3 \ln (t)$

Step 1.5

Simplify $- 4 \ln (z)$ by moving $4$ inside the logarithm.

$\ln (16) - \ln (x^{4}) + \ln (y^{0.5}) - \ln (z^{4}) + 6 \ln (w) + 3 \ln (t)$

Step 1.6

Simplify $6 \ln (w)$ by moving $6$ inside the logarithm.

$\ln (16) - \ln (x^{4}) + \ln (y^{0.5}) - \ln (z^{4}) + \ln (w^{6}) + 3 \ln (t)$

Step 1.7

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

$\ln (16) - \ln (x^{4}) + \ln (y^{0.5}) - \ln (z^{4}) + \ln (w^{6}) + \ln (t^{3})$

Step 2

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

$\ln (\frac{16}{x^{4}}) + \ln (y^{0.5}) - \ln (z^{4}) + \ln (w^{6}) + \ln (t^{3})$

Step 3

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

$\ln (\frac{16}{x^{4}} y^{0.5}) - \ln (z^{4}) + \ln (w^{6}) + \ln (t^{3})$

Step 4

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

$\ln (\frac{\frac{16}{x^{4}} y^{0.5}}{z^{4}}) + \ln (w^{6}) + \ln (t^{3})$

Step 5

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

$\ln (\frac{\frac{16}{x^{4}} y^{0.5}}{z^{4}} w^{6}) + \ln (t^{3})$

Step 6

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

$\ln (\frac{\frac{16}{x^{4}} y^{0.5}}{z^{4}} w^{6} t^{3})$

Step 7

Combine $\frac{16}{x^{4}}$ and $y^{0.5}$ .

$\ln (\frac{\frac{16 y^{0.5}}{x^{4}}}{z^{4}} w^{6} t^{3})$

Step 8

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{16 y^{0.5}}{x^{4}} \cdot \frac{1}{z^{4}} w^{6} t^{3})$

Step 9

Combine.

$\ln (\frac{16 y^{0.5} \cdot 1}{x^{4} z^{4}} w^{6} t^{3})$

Step 10

Multiply $16$ by $1$ .

$\ln (\frac{16 y^{0.5}}{x^{4} z^{4}} w^{6} t^{3})$

Step 11

Combine $\frac{16 y^{0.5}}{x^{4} z^{4}}$ and $w^{6}$ .

$\ln (\frac{16 y^{0.5} w^{6}}{x^{4} z^{4}} t^{3})$

Step 12

Combine $\frac{16 y^{0.5} w^{6}}{x^{4} z^{4}}$ and $t^{3}$ .

$\ln (\frac{16 y^{0.5} w^{6} t^{3}}{x^{4} z^{4}})$

Step 13

Reorder factors in $\ln (\frac{16 y^{0.5} w^{6} t^{3}}{x^{4} z^{4}})$ .

$\ln (\frac{16 w^{6} t^{3} y^{0.5}}{x^{4} z^{4}})$