# Name and Generalise This Linear Exponent Formula

In How to find the value of an unknown exponent?, user504882 provided an answer which inspired me to develop the following formula for solving exponential problems of the form $a^{k_1x + k_0} = b$ for $x$:

$$x = \log_a[(\frac{b}{a^{k_0}})^{\frac{1}{k_1}}].$$

On inspection, I see this can be simplified to

$$x = \frac{\log_a(b) - k_0}{k_1} ,$$

following well-known laws of logarithms. Does this formula have a name? Can it be generalised, say to exponents with rational quadratics or complex polynomials?

(If this sort of question is more approprate for the Math Exchange, feel free to migrate it.)

• If you simply take the $\log_a$ of both sides of the original equation, you get $k_1x+k_0=\log_a b$ directly. – dxiv Dec 15 '16 at 23:04
• smacks forehead Of course. I should have seen that from the beginning of the linked question. – S. G. Dec 16 '16 at 0:08