Finite Math Examples

\frac{d (\ln (k))}{d t} = \ln (A) - \frac{a}{R \cdot t}

$\frac{d (\ln (k))}{d t} = \ln (A) - \frac{a}{R \cdot t}$

Step 1

Simplify the expressions in the equation.

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Step 1.1

Simplify the left side.

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Step 1.1.1

Cancel the common factor of

d

$d$ .

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Step 1.1.1.1

Cancel the common factor.

\frac{d \ln (k)}{d t} = \ln (A) - \frac{a}{R \cdot t}

$\frac{d \ln (k)}{d t} = \ln (A) - \frac{a}{R \cdot t}$

Step 1.1.1.2

Rewrite the expression.

\frac{\ln (k)}{t} = \ln (A) - \frac{a}{R \cdot t}

$\frac{\ln (k)}{t} = \ln (A) - \frac{a}{R \cdot t}$

\frac{\ln (k)}{t} = \ln (A) - \frac{a}{R \cdot t}

$\frac{\ln (k)}{t} = \ln (A) - \frac{a}{R \cdot t}$

\frac{\ln (k)}{t} = \ln (A) - \frac{a}{R \cdot t}

$\frac{\ln (k)}{t} = \ln (A) - \frac{a}{R \cdot t}$

Step 1.2

Simplify the right side.

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Step 1.2.1

Multiply

R

$R$ by

t

$t$ .

\frac{\ln (k)}{t} = \ln (A) - \frac{a}{R t}

$\frac{\ln (k)}{t} = \ln (A) - \frac{a}{R t}$

\frac{\ln (k)}{t} = \ln (A) - \frac{a}{R t}

$\frac{\ln (k)}{t} = \ln (A) - \frac{a}{R t}$

\frac{\ln (k)}{t} = \ln (A) - \frac{a}{R t}

$\frac{\ln (k)}{t} = \ln (A) - \frac{a}{R t}$

Step 2

Multiply each term in

\frac{\ln (k)}{t} = \ln (A) - \frac{a}{R t}

$\frac{\ln (k)}{t} = \ln (A) - \frac{a}{R t}$ by

t R

$t R$ to eliminate the fractions.

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Step 2.1

Multiply each term in

\frac{\ln (k)}{t} = \ln (A) - \frac{a}{R t}

$\frac{\ln (k)}{t} = \ln (A) - \frac{a}{R t}$ by

t R

$t R$ .

\frac{\ln (k)}{t} (t R) = \ln (A) (t R) - \frac{a}{R t} (t R)

$\frac{\ln (k)}{t} (t R) = \ln (A) (t R) - \frac{a}{R t} (t R)$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of

t

$t$ .

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Step 2.2.1.1

Factor

t

$t$ out of

t R

$t R$ .

\frac{\ln (k)}{t} (t (R)) = \ln (A) (t R) - \frac{a}{R t} (t R)

$\frac{\ln (k)}{t} (t (R)) = \ln (A) (t R) - \frac{a}{R t} (t R)$

Step 2.2.1.2

Cancel the common factor.

\frac{\ln (k)}{t} (t R) = \ln (A) (t R) - \frac{a}{R t} (t R)

$\frac{\ln (k)}{t} (t R) = \ln (A) (t R) - \frac{a}{R t} (t R)$

Step 2.2.1.3

Rewrite the expression.

\ln (k) R = \ln (A) (t R) - \frac{a}{R t} (t R)

$\ln (k) R = \ln (A) (t R) - \frac{a}{R t} (t R)$

\ln (k) R = \ln (A) (t R) - \frac{a}{R t} (t R)

$\ln (k) R = \ln (A) (t R) - \frac{a}{R t} (t R)$

\ln (k) R = \ln (A) (t R) - \frac{a}{R t} (t R)

$\ln (k) R = \ln (A) (t R) - \frac{a}{R t} (t R)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of

t R

$t R$ .

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Step 2.3.1.1

Move the leading negative in

- \frac{a}{R t}

$- \frac{a}{R t}$ into the numerator.

\ln (k) R = \ln (A) t R + \frac{- a}{R t} (t R)

$\ln (k) R = \ln (A) t R + \frac{- a}{R t} (t R)$

Step 2.3.1.2

Factor

t R

$t R$ out of

R t

$R t$ .

\ln (k) R = \ln (A) t R + \frac{- a}{t R (1)} (t R)

$\ln (k) R = \ln (A) t R + \frac{- a}{t R (1)} (t R)$

Step 2.3.1.3

Cancel the common factor.

\ln (k) R = \ln (A) t R + \frac{- a}{t R \cdot 1} (t R)

$\ln (k) R = \ln (A) t R + \frac{- a}{t R \cdot 1} (t R)$

Step 2.3.1.4

Rewrite the expression.

\ln (k) R = \ln (A) t R - a

$\ln (k) R = \ln (A) t R - a$

\ln (k) R = \ln (A) t R - a

$\ln (k) R = \ln (A) t R - a$

\ln (k) R = \ln (A) t R - a

$\ln (k) R = \ln (A) t R - a$

\ln (k) R = \ln (A) t R - a

$\ln (k) R = \ln (A) t R - a$

Step 3

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

\ln (k) R - \ln (A) t R = - a

$\ln (k) R - \ln (A) t R = - a$

Step 4

Reorder factors in

\ln (k) R - \ln (A) t R

$\ln (k) R - \ln (A) t R$ .

R \ln (k) - t R \ln (A) = - a

$R \ln (k) - t R \ln (A) = - a$

Step 5

Add

t R \ln (A)

$t R \ln (A)$ to both sides of the equation.

R \ln (k) = - a + t R \ln (A)

$R \ln (k) = - a + t R \ln (A)$

Step 6

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

R \ln (k) - t R \ln (A) = - a

$R \ln (k) - t R \ln (A) = - a$