Solve for t natural log of k=-a/R*1/t+ natural log of A

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

Step 1

Rewrite the equation as $- \frac{a}{R} \cdot \frac{1}{t} + \ln (A) = \ln (k)$ .

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

Step 2

Simplify the left side.

Multiply $\frac{1}{t}$ by $\frac{a}{R}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, t R$

The LCM of one and any expression is the expression.

$t R$

Step 6

Multiply each term in $\ln (\frac{A}{k}) = \frac{a}{t R}$ by $t R$ to eliminate the fractions.

Multiply each term in $\ln (\frac{A}{k}) = \frac{a}{t R}$ by $t R$ .

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

Simplify the left side.

Reorder factors in $\ln (\frac{A}{k}) t R$ .

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

Simplify the right side.

Cancel the common factor of $t R$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 7

Divide each term in $t R \ln (\frac{A}{k}) = a$ by $R \ln (\frac{A}{k})$ and simplify.

Divide each term in $t R \ln (\frac{A}{k}) = a$ by $R \ln (\frac{A}{k})$ .

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

Simplify the left side.

Cancel the common factor of $R$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $\ln (\frac{A}{k})$ .

Cancel the common factor.

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

Divide $t$ by $1$ .

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

Solve for t natural log of k=-a/R*1/t+ natural log of A

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