One-to-one property of logs in EL3, EL5, and EL6

[AbbyANoble]

We define this property in Definition 5.6.7, and I think it would help to write it out in the notation instead of just words:

$log_b M = log_b N$ if and only if $M=N$.

Also, I would suggest changing the second half of the definition into an Observation. Something like "Notice this definition allows us to set the arguments (M and N) equal to each other once we get each side expressed as a single logarithm with matching bases."

Also, this seems to be the first time it's defined (in EL6), but it is mentioned in the previous section (EL5) as if it was defined in EL3. See

• Remark 5.5.1
• Activity 5.5.2(b)
• Activity 5.5.3(b)

We need to clear this up and figure out where we want everything located and referenced.

[siwelwerd]

I agree that is not a definition. And while I know it's common, I'd rather we stay away from the language "We can set them equal to each other" instead of saying precisely what we are doing, namely exponentiating both sides of the equation and simplifying.

We need to clear this up and figure out where we want everything located and referenced.

Maybe it fits nicely right after Remark 5.3.9 ? Then in Activity 5.6.8 we can still call back to it.

[AbbyANoble]

See #431 too, which is about Remark 5.3.9. I think that needs expanding.

At that point we haven't mentioned exponentiating both sides of an equation as a thing we can do, so I don't know that we should phrase it that way. Perhaps say we are applying the one-to-one property (or whatever it ends up being numbered, referencing it.)

Although, I actually couldn't find where we mentioned exponentiating both sides of an equation as a valid step. Nor could I find mention of the inverse property $b^{\log_b x} =x$. (I put this is #431.)

[siwelwerd]

At that point we haven't mentioned exponentiating both sides of an equation as a thing we can do

This isn't a new thing, it's just the idea we use all the time in solving equations that if a=b then f(a)=f(b) for whatever function f you like. Here f is exponentiation.

Nor could I find mention of the inverse property

I think my brain went ahead and filled this into Remark 5.3.9 as #431 suggests even though it isn't written there (yet?) 🤪

[StevenClontz]

This isn't a new thing, it's just the idea we use all the time in solving equations that if a=b then f(a)=f(b) for whatever function f you like. Here f is exponentiation.

I think an explicit reminder of this (or even having this as a Fact somewhere to reference, maybe in https://library.tbil.org/2024/precalculus/instructor/FN2.html ?) can't hurt, at least within the scaffolding of the first activity where they need to apply this fact.

[AbbyANoble]

Yeah. I don't think the new application (exponentiating) is something that they will recognize as following from other situations. It feels new at this level in the context of new functions. It should be mentioned.