Finite Math Examples

$\ln (\frac{60000}{20000}) e^{20 r}$

Step 1

Rewrite $\ln (\frac{60000}{20000})$ as $\ln (60000) - \ln (20000)$ .

$(\ln (60000) - \ln (20000)) e^{20 r}$

Step 2

Rewrite $\ln (20000)$ as $\ln (2^{5} \cdot 5^{4})$ .

$(\ln (60000) - \ln (2^{5} \cdot 5^{4})) e^{20 r}$

Step 3

Rewrite $\ln (2^{5} \cdot 5^{4})$ as $\ln (2^{5}) + \ln (5^{4})$ .

$(\ln (60000) - (\ln (2^{5}) + \ln (5^{4}))) e^{20 r}$

Step 4

Simplify each term.

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Step 4.1

Expand $\ln (2^{5})$ by moving $5$ outside the logarithm.

$(\ln (60000) - (5 \ln (2) + \ln (5^{4}))) e^{20 r}$

Step 4.2

Expand $\ln (5^{4})$ by moving $4$ outside the logarithm.

$(\ln (60000) - (5 \ln (2) + 4 \ln (5))) e^{20 r}$

Step 5

Apply the distributive property.

$(\ln (60000) - (5 \ln (2)) - (4 \ln (5))) e^{20 r}$

Step 6

Multiply $5$ by $- 1$ .

$(\ln (60000) - 5 \ln (2) - (4 \ln (5))) e^{20 r}$

Step 7

Multiply $4$ by $- 1$ .

$(\ln (60000) - 5 \ln (2) - 4 \ln (5)) e^{20 r}$

Step 8

Simplify each term.

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Step 8.1

Rewrite $\ln (60000)$ as $\ln (2^{5} \cdot 3 \cdot 5^{4})$ .

$(\ln (2^{5} \cdot 3 \cdot 5^{4}) - 5 \ln (2) - 4 \ln (5)) e^{20 r}$

Step 8.2

Rewrite $\ln (2^{5} \cdot 3 \cdot 5^{4})$ as $\ln (2^{5} \cdot 3) + \ln (5^{4})$ .

$(\ln (2^{5} \cdot 3) + \ln (5^{4}) - 5 \ln (2) - 4 \ln (5)) e^{20 r}$

Step 8.3

Simplify each term.

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Step 8.3.1

Rewrite $\ln (2^{5} \cdot 3)$ as $\ln (2^{5}) + \ln (3)$ .

$(\ln (2^{5}) + \ln (3) + \ln (5^{4}) - 5 \ln (2) - 4 \ln (5)) e^{20 r}$

Step 8.3.2

Expand $\ln (2^{5})$ by moving $5$ outside the logarithm.

$(5 \ln (2) + \ln (3) + \ln (5^{4}) - 5 \ln (2) - 4 \ln (5)) e^{20 r}$

Step 8.3.3

Expand $\ln (5^{4})$ by moving $4$ outside the logarithm.

$(5 \ln (2) + \ln (3) + 4 \ln (5) - 5 \ln (2) - 4 \ln (5)) e^{20 r}$

Step 9

Subtract $5 \ln (2)$ from $5 \ln (2)$ .

$(\ln (3) + 4 \ln (5) + 0 - 4 \ln (5)) e^{20 r}$

Step 10

Add $\ln (3)$ and $0$ .

$(4 \ln (5) + \ln (3) - 4 \ln (5)) e^{20 r}$

Step 11

Subtract $4 \ln (5)$ from $4 \ln (5)$ .

$(0 + \ln (3)) e^{20 r}$

Step 12

Add $0$ and $\ln (3)$ .

$\ln (3) e^{20 r}$

Step 13

Reorder factors in $\ln (3) e^{20 r}$ .

$e^{20 r} \ln (3)$