I need to solve for x but x is a power

Question

The original question was:

3^2x-4(3^x)+3=0

I 'u' subbed for 3^xand got

u²-4u+3 = 0

(u-1)(u-3) = 0

therefore

u = 1 or 3 (null factor theorem)

therefore

3^x=1

or

3^x=3

I don't know what to do next though.

Answer 1

Hi William,

I realize this is an old question, so I hope you were able to find the answer at the time or get help from your teacher. I might as well answer it anyway, though.

Your substitution method is clever, I have never seen these problems done that way but it's perfectly valid.

The question you arrived at was which values of x would satisfy the equations 3^x = 1 & 3^x = 3.

It is actually possible to find these values with a little general knowledge about exponents.

Here are two general facts about exponents:

Fact 1: a¹ = a, for all real numbers a.

So 3¹ = 3, and x = 1 is our solution to that factor.

Fact 2: a⁰ = 1, for all real numbers a where a ≠ 0.

(And don't get me started on 0⁰, that's a whole other can of worms!)

So, 3⁰ = 1, and our solution to that factor x = 0.

So the x-intercepts for this function would therefore be (0,0) & (1,0).

Now, in the general case, the mathematical tool we use to solve problems like this is called logarithms. This is how I would have approached the problem.

Logarithms are like the opposite of exponents. We write them like this: log_a(c), meaning, "the log base a of c," where a and c are whatever numbers you may be dealing with.

And the way they work is that, if a^b = c, then by definition, log_a(c) = b.

In other words, the mathematical expression log_a(c) is equivalent to asking: "a to the power of what is equal to c?"

So in this case, we could write log₃(1), and we would find that log₃(1) is equal to 0.

And similarly, we would find that log₃(3) = 1.

If we were to use a TI calculator to solve this, we would also need to utilize something called the change of base formula which is as follows:

log_a(c) = log_b(c) ÷ log_b(a), where b is just any positive number you can think of. This formula holds true for all positive numbers a, b, and c as long as none of them are equal to one and like I say, they are all positive numbers.

The reason you would need to know the change of base formula to use most TI calculators to solve logarithms is that most TI calculators only have 2 logarithm buttons, "ln" and "log". And "log" is the log base 10 while "ln" is the log base e or Euler's constant. So if you want to calculate a logarithm with some other base you'd need to use the change of base formula to do it.

Anyway, logarithms are useful for situations where it's not so obvious what the answer might be, such as if we were talking about say the log base 3 of 2, which would be just some decimal. But this problem you can actually solve without logarithms if you leverage the facts I mentioned above.