Solve exponential equation

Question

I'm dealing with a problem here. I'm trying to solve this exponential equation but I cannot find the solution: $$3^{x-1} + 3^{x-2} + 3^{x-3} + 3^{x-4}\cdot3^{x-5} + 3^{x-6}=364$$

Can anyone please tell me what to do ?

Thank you!

Answer 1

Another approach from the one Will suggested is this: recognize that the left hand side is a geometric series, namely $$ 3^{x-6} (1 + 3 + 3^{2} + ... + 3^{5}) $$ whose sum is $$ 3^{x-6} \cdot \frac{1 - 3^{6}}{1 - 3}. $$ Now the number $x$ appears only once, and you can do some algebra to get to a solution.

Answer 2

Write $364 = 3\cdot 11^2+1 = 3\cdot (3^2+2)^2+3^0$. Play around with expanding the quantity $3\cdot (3^2+2)^2$ and converting all terms to a power of $3$. This is tantamount to writing $364$ into base-$3$ and will immediately tell you if there is an $x$ that satisfies your equation. If you do this and don't get a basis representation that looks like your equation, then no solution exists because representations are unique.

Answer 3

Multiplying the original equation by $3$:

$$3^x+3^{x-1}+3^{x-2}+3^{x-3}+3^{x-4}+3^{x-5}=1092$$

Now subtracting the original equation from this:

$$3^x-3^{x-6}=728$$ $$3^x-\frac{3^x}{729}=\frac{728}{729}3^x=728$$ $$3^x=729,x=6$$

Update: Now that the multiplication has been brought to my attention, it appears John's solution is best. I wish I could see some sort of algebra trick to employ to at least help factor anything, but I don't. It's just an ugly quadratic probably best suited to the quadratic formula.

Answer 4

This equation can be re-written as:

$$\frac{3^x}{3}+ \frac{3^x}{9} + \frac{3^x}{27}+ \left(\frac{3^x}{81}\times \frac{3^x}{243}\right) + \frac{3^x}{729} = 364$$

let $3^x = a$ and multiply through by 19683.

$$ a^2 + 9504a -7164612 = 0$$

$$(a-702)(a+10206)= 0$$

$$a = 702 \ \text{or}\ a = -10206$$

Hence: $$3^x = 702$$

Taking $\ln$ of both sides:

$$x = \frac{\ln702}{\ln3} $$

Answer 5

$$ 364=3^x \sum_{j=1}^6 3^{-j} = 3^{x-1} \sum_{j=0}^5 3^{-j} = 3^{x-1} \frac{1-3^{-6}}{1-3^{-1}}=3^x\frac{364}{3^6} $$ so $x=6$