Is it possible to isolate n in this equation (log related) and how?

Question

Here is the equation:

$$ 3^{2n+2} - 2^{n+1} = 77 $$

The answer is $n = 1$, but is it possible to isolate $n$ and find it mathematically? I am unable to do so, despite being familiar with $\log$ properties.

Answer 1

As the answer to your question "Is it possible to isolate n and find it mathematically?" - no, it isn't possible for arbitrary $n$. If $n$ is natural number, then you can do some divisibility tests to determine for which $n$ the equality is true. But, you cannot apply logarithm here.

You can solve equations of the form

$$a^x+bx+c=0$$

for $x$ where $a,b,c$ are parameters, but your equation cannot be reduced to that form.

Answer 2

Yes it is possible, but not in a standard way:

For $a\gt b\ge 0$ define the function $g_{a,b}(x)=a^x-b^x$ on non-negative reals. This is well defined and infinitely differentiable. $\dfrac{d}{dx}(g_{a,b}(x))=a^x\ln a-b^x\ln b=\ln\left(\dfrac{a^{a^x}}{b^{b^x}}\right).$ The derivative is always positive and therefore $g_{a,b}$ is increasing on its domain. Hence $g_{a,b}^{-1}$ is a well defined function and here we write $n$ as $g_{9,2}^{-1}(77)-1.$

Answer 3

Let us make the problem more general and, using algebra, consider that you look for the zero of function $$f(x)=(a+1)^{2x} - a^{x} -c$$ where $a>0$, $c>0$ and $x=n+1$.

This can rewrite $$g(x)=2x\log(a+1)-\log(a^x+c)$$ which almost looks as a straight line because of the logarithmic transform.

Suppose we use, as in your problem, $a=3$, $b=2$ and $c=531377$; plot function $g(x)$ and you will see that the solution is $x=6$ corresponding to $n=5$.