Implies or iff between two equations that are the same?

Question

Say I have the following equality $$ \ln x = d $$ This means that $x = e^d$. However, I am questioning whether, when writing this equivalence in one go, whether to use $\Rightarrow$(implies) or $\Leftrightarrow$(iff). Would you write $$ \ln x = d \Rightarrow x = e^d $$ or $$ \ln x = d \Leftrightarrow x = e^d $$ and why -- I am of course interested in the general case with a possibly longer sequence of equivalent expressions? Where my doubt is primarily coming from is that both are correct but I have seen a couple textbooks using $\Rightarrow$ while $\Leftrightarrow$ holds..

Answer 1

It is true that $\ln(x) = d$ $\Leftrightarrow$ $x = e^d$.

However, usually when we teach, we often teach exponentiation before we teach logarithms. In this case, when we define logarithms as the inverse function of exponentiation, we then say

Definition. We say that $\ln(x) = d$ if $x = e^d$.

Symbolically, this definition would look like: $\ln(x) = d$ $\Leftarrow$ $x = e^d$.

Usually, when we make a definition, even though the clause that the definiendum (the thing being defined) is a shorthand for must hold if and only if the definiendum would hold, yet it is customary to only use the participle "if" rather than the precise "if and only if". For example, the proper way to make the above definition would actually be to say "We say that $\ln(x) = d$ if and only if $x=e^d$." However, due to our custom, we drop the "and only if" participle.

In my above context, if the natural logarithm was defined first before exponentiation, (for example by saying that $\ln'(x) = 1/x$ and $\ln(1) = 0$), then the exponential function could then be defined as its inverse function, so

Definition. We say that $x = e^d$ if $\ln(x) = d$.

Symbolically, $x= e^d$ $\Rightarrow$ $\ln(x) = d$, or equivalently, $\ln(x) = d \implies x = e^d$.

Answer 2

While ⟺ is correct in the given example, frequently—due to the additional time and cognitive load required to verify the ⟸ direction—it is easy to overlook the fact that ⟹ is correct but ⟺ is actually not.

If there is no particular need to indicate the ⟸ direction, ⟹ similarly makes for a smoother reading experience than ⟺.

When solving equations, ⟺ has an advantage over ⟹ in assuring us that the solution set and candidate solution set are the same, but even so, using ⟺ as the default symbol can be prone to autopilot carelessness and not noticing extraneous solutions.

For example, in the quoted solution here, it may feel natural to insert ⟺ between every step, and be easy to miss that ⟸ is actually incorrect for step $(\#).$