I'm trying to solve the following for b:
$\ln ab^t = -0.12t + 4.67$ where $-0.12t + 4.67$ is the equation of a straight line.
I apply basic log rules:
$\ln a + t \ln b = -0.12t + 4.67$
According to the mark scheme / answer, you can just compare the coefficients of 't':
$-0.12 = \ln b$
But I don't understand, doesn't that imply the following:
$4.67 = \ln a$
If so, how can we be sure when there are two constants, a and b?
using the logarithm rules we get $$\ln(a)+t\ln(b)=-\frac{3}{25}t+\frac{467}{100}$$ and this is equivalent to $$\ln(b)=\frac{1}{t}\left(-\frac{3}{25}t+\frac{467}{100}-\ln(a)\right)$$ thus $$b=e^{\frac{1}{t}\left(-\frac{3}{25}t+\frac{467}{100}-\ln(a)\right)}$$