solve for something else

• Apr 15th 2014, 10:53 AM
leviriven
solve for something else
I frequently use the following two formulas in my work.

(1) V = A / (1 + kd)
(2) V = Ae^(-kd)

These formulas provide alternative methods for finding the value V, when the variables A, k, and d are known. However, currently I have the values for V, d, and A, but not for k. I would like to reformulate these expressions to solve for k.

I have completely forgotten the necessary algebraic rules. I would greatly appreciate if someone can help me reformulate these equations.
• Apr 15th 2014, 11:55 AM
earboth
Re: solve for something else
Quote:

Originally Posted by leviriven
I frequently use the following two formulas in my work.

(1) V = A / (1 + kd)
(2) V = Ae^(-kd)

These formulas provide alternative methods for finding the value V, when the variables A, k, and d are known. However, currently I have the values for V, d, and A, but not for k. I would like to reformulate these expressions to solve for k.

I have completely forgotten the necessary algebraic rules. I would greatly appreciate if someone can help me reformulate these equations.

Helo,

to #1:

$\displaystyle V = \frac{A }{1 + k\cdot d}~\implies~V(1+k\cdot d)=A~\implies~V+V \cdot d \cdot k = A\\ \\ \implies~V \cdot d \cdot k = A-V~\implies~k=\frac{A-V}{V \cdot d }$

to #2:

$\displaystyle V = A \cdot e^{-k\cdot d}~\implies~\frac VA=e^{-k\cdot d}$ Now take the natural logarithm on both sides:

$\displaystyle \ln\left(\frac VA \right)= - k \cdot d~\implies~k = -\frac1d \cdot (\ln(V) - \ln(A))$
• Apr 15th 2014, 12:03 PM
Plato
Re: solve for something else
Quote:

Originally Posted by leviriven
I frequently use the following two formulas in my work.
(1) V = A / (1 + kd)
(2) V = Ae^(-kd).

(1) $k=\dfrac{A-V}{dV}$

Assume all variables are positive and use the natural logarithm:

(2) $k=\dfrac{\log(A)-\log(V)}{d}$
• Apr 15th 2014, 12:06 PM
JeffM
Re: solve for something else
Quote:

Originally Posted by leviriven
I frequently use the following two formulas in my work.

(1) V = A / (1 + kd)
(2) V = Ae^(-kd)

These formulas provide alternative methods for finding the value V, when the variables A, k, and d are known. However, currently I have the values for V, d, and A, but not for k. I would like to reformulate these expressions to solve for k.

I have completely forgotten the necessary algebraic rules. I would greatly appreciate if someone can help me reformulate these equations.

Equation I: $d \ne 0\ and\ V = \dfrac{A}{1 + kd} \implies V + kdV = A \implies kdV = A - V \implies k = \dfrac{A - V}{dV}.$

In the next equation, I am assuming that e is Euler's number, a constant, rather than a variable.

Equation II: $d \ne 0\ and\ V = Ae^{(-kd)} \implies ln(V) = ln\left(Ae^{(-kd)}\right) = ln(A) - kd *ln(e) = ln(A) * kd \implies$

$kd = ln(A) - ln(V)= ln\left(\dfrac{A}{V}\right) \implies k = \dfrac{1}{d} * ln\left(\dfrac{A}{V}\right) =ln\left(\left\{\dfrac{A}{V}\right\}^{(1/d)}\right) \implies k = ln\left(\sqrt[d]{\dfrac{A}{V}}\right).$

EDIT: I see others posted while I was writing. The three results for Equation II compute to the same answer.
• Apr 15th 2014, 12:35 PM
leviriven
Re: solve for something else
Thank you very much! This looks great!