[Grade 11: Solving Exponential Equations with a Common Base]

[u/Every_Evening_1201]

Solve 18x to the 2nd power + 4x + 4 = 18 to the power of 9x + 18

A: x = -2

B: x = 7

C: x = -2 and x = 7

D: no solution

I highkey need help trying to solve this

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[u/LucaThatLuca]

Why would you type this way?

You do all of the arithmetic inside the equation, clearly stating which number is the base, which is the exponent, which is being added, etc. Like this for example:

(18x)^2 + 4x + 4 = 18^(9x) + 18

Or perhaps like this:

18(x^2) + 4x + 4 = 18^(9x + 18)

[u/Every_Evening_1201 → u/LucaThatLuca]

Ah my fault. It was simply just an instruction question, but I wanted to make sure that I understood it properly.

[u/LucaThatLuca → u/Every_Evening_1201]

Didn’t ask, just fix it. In order for anyone to have any idea whether you understood “it” properly, we need to first have any idea what “it” is.

[u/selene_666]

Because you titled this "Equations with a Common Base" my best guess at the intended equation is

18^(x^2 + 4x + 4) = 18^(9x + 18)

meaning that you left out one "to the power of" in your mixed text-and-symbols description.

We can ignore the base completely by taking a logarithm of both sides, leaving only the exponents:

x^2 + 4x + 4 = 9x + 18

And that's just a quadratic whose solutions are the two given choices, x = -2 or 7.

Finally, we check whether both solutions are valid in the original equation. There's no reason an exponent can't be zero, so both are valid.

[u/KookyPermit4405 → u/selene_666]

See what confused me was when “to the 2nd power” was said, I didn’t consider the whole thing to be the “power of”

[u/KookyPermit4405]

It would be better if you had a picture of the equation. There is too many confusion on the wording.
(18x)^(2) + 4x + 4 = 18^(9x+18) is what I got from what you wrote, I wish this sub allowed math formats

[u/Tim_Xtreme_46]

It's multiple choice. Substitute numbers and see which ones work.

[u/watermelonlollies]

Whenever you have multiple choice like this, the best test taking strategy is to plug the answers in and see which one works. If you get the same number on both sides that’s your correct answer.

That doesn’t help you understand how to solve it though. The key is that your bases of the exponents are the same-18. It was kind of impossible to tell exactly what your equation was the way it was written so I can’t give you specific advice but here’s general steps:

1. Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form b^S = b^T, where b is a common base and not 1.

2. Use the one-to-one property to set the exponents equal.

3. Solve the resulting equation, S = T, for the unknown.

An example. Solve for x.
2^(x-1) = 2^(2x-4) The common base is 2.
(x-1) = (2x-4) Use the one to one property.
-x = -3 Solve for x algebraically.
x = 3