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.)

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

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