How to rearrange the given equation for Ao: ln (Ao/A) = k*t
Have you forgotten your Algebra I?
Ao/A = kt
Ao = Akt
No but what about the ln (natural log)??
the equation is [ln(Ao/A)=kt]
This is from the 1st order integrated rate law => A = Aoe^-kt
solve for Ao = A/e^-kt =Ae^kt
(I showed solution for A in earlier post from (ln) function.)
To rearrange the given equation ln(Ao/A) = k*t for Ao, you can follow these steps:
1. Start by multiplying both sides of the equation by -1. This will change the sign of ln(Ao/A) to -ln(Ao/A), and k*t to -k*t. The equation becomes -ln(Ao/A) = -k*t.
2. Use the logarithmic property that ln(A/B) = ln(A) - ln(B). Apply this property to the left side of the equation. The equation now becomes ln(A) - ln(Ao) = -k*t.
3. Rearrange the terms by adding ln(Ao) to both sides of the equation. The equation becomes ln(A) = ln(Ao) - k*t.
4. Finally, exponentiate both sides of the equation by the base e, which is the inverse function of the natural logarithm ln(x). This will eliminate the ln function on the left side. The equation becomes e^(ln(A)) = e^(ln(Ao) - k*t).
5. Simplify the equation by applying the exponential property e^(ln(x)) = x to both sides. This results in A = Ao * e^(-k*t).
Therefore, the rearranged equation for Ao is Ao = A / (e^(-k*t)).