A grammar question on taking powers

Question

"... we find that

$$ 3x>2x .$$

raising both sides as a power of $2$, we get

$$ 2^{3x}>2^{2x} .$$

So the inequa..."

My question, is the part "raising both sides as a power of $2$" suitable for the formal(I mean in a mathematics book or related materials like articles) mathematical language?

Edit: The way I ask tis question is a little bit confusing I think. The mathematical equtions between thefirst quatition marks was actually random. What I want was is "raising both sides as a power of $2$" grammatically true? if it is not, then what would you say?

Anyway, your answers are quite helpful for the random question :)

Answer 1

• "Do $2$ to the power of.... on both sides." (is not correct in terms of grammar, but is most easily understood IMO)

• "Put both sides as a power of $2$"

• "Make $2$ as a base of $2^x$ and $3^x$" (although this might be better for logarithms)

• "Exponentiate with respect to base $2$"

• "Raise both sides in terms of base $2$"

Otherwise, raising both sides to the power $2$ is OK.

Answer 2

The function $f(x)=2^x$ is a strictly increasing function. Hence, if $y<z$, then $f(y)<f(z)$.

In the OP, $y=2x$ while $z=3x$. Note that this holds only for $x>0$.

In therms of the phrasing, it is a bit awkward but correct. In my view, this phrasing is not significantly worse than other awkward alternatives.

Answer 3

"Raising both sides as a power of 2" is fine.

Note that the step is correct in this case since $f(x)=2^x$ is a strictly increasing function when $3x>2x$ that is $x>0$.

Answer 4

This is more an English-language question than a mathematical question. The phrase "raising both sides as a power of $2$" sounds ungrammatical to me. Indeed, Merriam-Webster does not list "raise as" as a possibility, but it does list the mathematical notion "raise to" (https://www.merriam-webster.com/dictionary/raise).

I would rephrase it as "Exponentiating both sides" (ignoring the base $2$), or even just "Therefore" (trusting that the reader will see that both sides have been exponentiated). I feel that "Exponentiate with base $2$" and similar phrasing is both unnatural and too verbose.