How do you solve this exponential equation?

Question

$3(16)^x+2(81)^x=5(36)^x$

How do you change the bases to combine the terms? The correct answer should be 0 and 0.5.

Edit: So this equation can't be solved algebraically? I have to use creative logic to solve it?

Answer 1

Let $a=4^x$ and $b=9^x$. Note that our equation can be rewritten as $$(3a-2b)(a-b)=0.$$

Remark: Minor modification of the numbers will give us an equation whose solutions do not have nice closed form.

Answer 2

It might be useful if you can see this as

$3.(2^{x})^4+2.(3^x)^4=(2+3)(2^x.3^x)$

Suppose if at all $2^x=3^x=1$ then you have $2+3=2+3$.. and what does this mean for $x$?

Answer 3

Note that $$3(16)^x=3(4)^{2x}, 2(81)^x=2(9)^{2x}, \text{and}\; 5(36)^x=5(6)^{2x}$$ If $2x=1$, you get that $12+18=30$, a true statement. Thus, $x=\frac{1}{2}$. The case where $x=0$ is trivial.

Answer 4

Hint: exponentiate both sides of the equation using the function $e^{x}$ and then simlify.