How can I convert a cubic to a linear equation using logs or other method?

Question

is it possible and if so how, to convert a cubic into a linear equation?

$$Y=-0.152x^3+5.79x^2-86.8x+719$$

Is there anyway, such as using logs, to convert it into a Linear form?

Cheers

Jack

(someone had some similar maths at How can I convert fifth order polynomial to a linear equation using logs?)

(Anyway to work it into excel would be incredible!!! cheers)

Answer 1

First, find the three roots of the polynomial $\;Y=-0.152x^3+5.79x^2-86.8x+719$.

These are $x_1=21.871,\;x_2=8.11057 + 12.2678 i,\;x_3=8.11057 - 12.2678 i$, so we can write the polynomial as

$$Y=\left(X-21.871\right)\left(X-(8.11057\pm 12.2678 i)\right)$$

Now taking $\ln$ on both sides we get

$$\ln(Y)=\ln\left(X-21.871\right)+\ln\left(X-(8.11057 + 12.2678 i)\right)+\ln\left(X-(8.11057 - 12.2678 i)\right)$$

This is your new equation obtained by taking logarithms.