isolating a exponent from the sum of two terms ($16^x - 10^x = y$)

Question

$16^x - 10^x = y$ How can I isolate x in this case? Not much other information to give, This is just an equation that I came up with, whilst messing around.

Answer 1

You... can't.

What you wrote down is an equation that has no closed form solution. Most equations of the type $$a^x + b^x = c$$ fall into that category, meaning that in general, there is no other way to neatly (i.e. using trigonometric functions, exponentials, logarithms, fractions and so on) describe what $x$ is other than "the solution of that equation over there."

Now, not, I am neither saying "there is no solution", nor "there is never a simple solution".

• For example, in your case, you know that $\lim_{x\to\infty} 10^x - 6^x = \infty$ and you can show that, at least for positive values of $x$, the function is injective. So, if $y>0$, you can be certain there is precisely one solution to your equation, even though you don't know what it is.
• There are values for $y$ where you can find the solution quickly and in closed form. For example, if $y=0$, then the solution is $x=0$, and if $y=6$, then the solution is $x=1$. And for $y=4-\sqrt 10$, the solution is $x=\frac12$. And so on.