Basic Math Examples

$241 \div 4000 = 241 \div 2^{m} \cdot 5^{n}$

Step 1

Rewrite the equation as

$241 \div 2^{m} \cdot 5^{n} = 241 \div 4000$ .

$241 \div 2^{m} \cdot 5^{n} = 241 \div 4000$

Step 2

Simplify

$241 \div 2^{m} \cdot 5^{n}$ .

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Step 2.1

Rewrite the division as a fraction.

$\frac{241}{2^{m}} \cdot 5^{n} = 241 \div 4000$

Step 2.2

Combine

$\frac{241}{2^{m}}$ and

$5^{n}$ .

$\frac{241 \cdot 5^{n}}{2^{m}} = 241 \div 4000$

Step 3

Rewrite the division as a fraction.

$\frac{241 \cdot 5^{n}}{2^{m}} = \frac{241}{4000}$

Step 4

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (\frac{241 \cdot 5^{n}}{2^{m}}) = \ln (\frac{241}{4000})$

Step 5

Expand the left side.

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Step 5.1

Rewrite

$\ln (\frac{241 \cdot 5^{n}}{2^{m}})$ as

$\ln (241 \cdot 5^{n}) - \ln (2^{m})$ .

$\ln (241 \cdot 5^{n}) - \ln (2^{m}) = \ln (\frac{241}{4000})$

Step 5.2

Expand

$\ln (2^{m})$ by moving

$m$ outside the logarithm.

$\ln (241 \cdot 5^{n}) - (m \ln (2)) = \ln (\frac{241}{4000})$

Step 5.3

Remove parentheses.

$\ln (241 \cdot 5^{n}) - m \ln (2) = \ln (\frac{241}{4000})$

Step 5.4

Rewrite

$\ln (241 \cdot 5^{n})$ as

$\ln (241) + \ln (5^{n})$ .

$\ln (241) + \ln (5^{n}) - m \ln (2) = \ln (\frac{241}{4000})$

Step 5.5

Expand

$\ln (5^{n})$ by moving

$n$ outside the logarithm.

$\ln (241) + n \ln (5) - m \ln (2) = \ln (\frac{241}{4000})$

Step 6

Simplify the left side.

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Step 6.1

Move

$\ln (241)$ .

$n \ln (5) - m \ln (2) + \ln (241) = \ln (\frac{241}{4000})$

Step 6.2

Reorder

$n \ln (5)$ and

$- m \ln (2)$ .

$- m \ln (2) + n \ln (5) + \ln (241) = \ln (\frac{241}{4000})$

Step 7

Move all the terms containing a logarithm to the left side of the equation.

$- m \ln (2) + n \ln (5) + \ln (241) - \ln (\frac{241}{4000}) = 0$

Step 8

Use the quotient property of logarithms,

$\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$- m \ln (2) + n \ln (5) + \ln (\frac{241}{\frac{241}{4000}}) = 0$

Step 9

Simplify each term.

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Step 9.1

Multiply the numerator by the reciprocal of the denominator.

$- m \ln (2) + n \ln (5) + \ln (241 (\frac{4000}{241})) = 0$

Step 9.2

Cancel the common factor of

$241$ .

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Step 9.2.1

Cancel the common factor.

$- m \ln (2) + n \ln (5) + \ln (241 (\frac{4000}{241})) = 0$

Step 9.2.2

Rewrite the expression.

$- m \ln (2) + n \ln (5) + \ln (4000) = 0$

Step 10

Move all terms not containing

$m$ to the right side of the equation.

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Step 10.1

Subtract

$n \ln (5)$ from both sides of the equation.

$- m \ln (2) + \ln (4000) = - n \ln (5)$

Step 10.2

Subtract

$\ln (4000)$ from both sides of the equation.

$- m \ln (2) = - n \ln (5) - \ln (4000)$

Step 11

Divide each term in

$- m \ln (2) = - n \ln (5) - \ln (4000)$ by

$- \ln (2)$ and simplify.

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Step 11.1

Divide each term in

$- m \ln (2) = - n \ln (5) - \ln (4000)$ by

$- \ln (2)$ .

$\frac{- m \ln (2)}{- \ln (2)} = \frac{- n \ln (5)}{- \ln (2)} + \frac{- \ln (4000)}{- \ln (2)}$

Step 11.2

Simplify the left side.

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Step 11.2.1

Dividing two negative values results in a positive value.

$\frac{m \ln (2)}{\ln (2)} = \frac{- n \ln (5)}{- \ln (2)} + \frac{- \ln (4000)}{- \ln (2)}$

Step 11.2.2

Cancel the common factor of

$\ln (2)$ .

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Step 11.2.2.1

Cancel the common factor.

$\frac{m \ln (2)}{\ln (2)} = \frac{- n \ln (5)}{- \ln (2)} + \frac{- \ln (4000)}{- \ln (2)}$

Step 11.2.2.2

Divide

$m$ by

$1$ .

$m = \frac{- n \ln (5)}{- \ln (2)} + \frac{- \ln (4000)}{- \ln (2)}$

Step 11.3

Simplify the right side.

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Step 11.3.1

Simplify each term.

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Step 11.3.1.1

Dividing two negative values results in a positive value.

$m = \frac{n \ln (5)}{\ln (2)} + \frac{- \ln (4000)}{- \ln (2)}$

Step 11.3.1.2

Dividing two negative values results in a positive value.

$m = \frac{n \ln (5)}{\ln (2)} + \frac{\ln (4000)}{\ln (2)}$