Solve for x: 2^3+2^3+2^3+2^3=2^x

[LogarithmLuke]

Homework Statement

Solve the following equation: 2^3+2^3+2^3+2^3=2^x[/B]

Homework Equations

log(a)^x=x*log(a)[/B]

The Attempt at a Solution

What i attempted was to log both sides, bring down the exponents, and summarize them. This left me with 12*log(2)=x*log(2). I then divide both sides by log(2) and get x=12, which is wrong. Please note that all of this was with logarithms with base number 10.

 

[BvU]

Hi,

Where did you get your relevant equation ? It's dead (*) wrong, but visually close to the right one: $\log (a^x) = x \log a$.

Taking logarithms of a sum is generally a bad idea. For a product you are better off. So make a product of the lefthand side.
And look at that. Maybe you don't need to take logarithms, but you might be able to use a simpler equation about exponentiation.

PS the term "summarizing" isn't all that mathematically sound... "sum" is what you mean. But you can't do that.

(*) [edit] well, dangerous is a better expression. It raises confusion between $(\log a)^x$ and $(\log (a^x))$

 

[HallsofIvy]

I can see no reason to use logarithms. You are just trying to find x such that $2^x= 32$. That should be elementary.

 

[RUber]

I second what BvU says. If this is an exercise of the properties of exponents, you should notice that 2^3 is a repeated sum...how many times is it repeated?
Rewrite it as a coefficient times 2^3, then write your coefficient as a power of 2.
Now, you will be able to use properties of exponents to quickly solve for x.

Otherwise, just add 8+8+8+8 and do what HallsofIvy suggested.

And if you are really feeling like using logarithms ... use log base 2.