Precalculus Examples

$\ln (10 t^{2}) u v^{- 3}$

Step 1

Rewrite $\ln (10 t^{2})$ as $\ln (10) + \ln (t^{2})$ .

$(\ln (10) + \ln (t^{2})) u v^{- 3}$

Step 2

Expand $\ln (t^{2})$ by moving $2$ outside the logarithm.

$(\ln (10) + 2 \ln (t)) u v^{- 3}$

Step 3

Simplify each term.

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Rewrite $\ln (10)$ as $\ln (2 (5))$ .

$(\ln (2 (5)) + 2 \ln (t)) u v^{- 3}$

Rewrite $\ln (2 (5))$ as $\ln (2) + \ln (5)$ .

$(\ln (2) + \ln (5) + 2 \ln (t)) u v^{- 3}$

Step 4

Apply the distributive property.

$(\ln (2) u + \ln (5) u + 2 \ln (t) u) v^{- 3}$

Step 5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(\ln (2) u + \ln (5) u + 2 \ln (t) u) \frac{1}{v^{3}}$

Step 6

Multiply $\ln (2) u + \ln (5) u + 2 \ln (t) u$ by $\frac{1}{v^{3}}$ .

$\frac{\ln (2) u + \ln (5) u + 2 \ln (t) u}{v^{3}}$

Step 7

Factor $u$ out of $\ln (2) u + \ln (5) u + 2 \ln (t) u$ .

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Factor $u$ out of $\ln (2) u$ .

$\frac{u \ln (2) + \ln (5) u + 2 \ln (t) u}{v^{3}}$

Factor $u$ out of $\ln (5) u$ .

$\frac{u \ln (2) + u \ln (5) + 2 \ln (t) u}{v^{3}}$

Factor $u$ out of $2 \ln (t) u$ .

$\frac{u \ln (2) + u \ln (5) + u (2 \ln (t))}{v^{3}}$

Factor $u$ out of $u \ln (2) + u \ln (5)$ .

$\frac{u (\ln (2) + \ln (5)) + u (2 \ln (t))}{v^{3}}$

Factor $u$ out of $u (\ln (2) + \ln (5)) + u (2 \ln (t))$ .

$\frac{u (\ln (2) + \ln (5) + 2 \ln (t))}{v^{3}}$