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If we have an equation we want to solve such as $\sqrt{x} = 3$, we can say something such as square both sides or "raise both sides to the power of 2", to arrive at $x = 9$. So $3 \rightarrow 3^2$ with this action.

Is there any standard terminology which can be used for taking a number (or each side of an equation) and using it as the exponent of some base like $2 \rightarrow 3^2$?

For example, consider the following equation and solution: \begin{align}\log_5 x & = 3 \\ 5^{\log_5 x} & = 5^{3} \tag{*}\label{} \\ x & = 125 \end{align}

How should I describe the step $(*)$ labeled here? The way I currently use to describe something like this is "take each side as an exponent of base 5".

I suppose I could say something like "raise 5 to the power of both sides", but I want to avoid this as the direct object of the sentence is not the thing I am starting with. That is, conceptually I am not doing any action to the number $5$ - rather, I am doing something to the numbers $\log_5 x$ and $3$ - so these should be the direct object(s) of the sentence.

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I would say "exponentiation". For example, going from $a=b$ to $e^a=e^b$, I would say that I am "exponentiating" both sides.

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  • $\begingroup$ This is also meaningful when we differentiate between "powertowers" and "exponentialtowers" when we talk about tetration, aka "iterated exponentiation" $\endgroup$
    – Gottfried Helms
    Commented Mar 23, 2016 at 13:02

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