Are the notations used for exponents and logarithms confusing?

Consider multiplication. We express 2+2+2 as 3*2, which seems quite natural

I completely disagree. I consider 3*2 to mean 3 + 3, not 2+2+2. I don’t consider that backwards at all. It’s saying “The number 3, added two times”.

Correspondingly, 3^2 means 3 * 3, reading “The number 3, multiplied two times”.

Interesting question that I am not sure I can answer other than from my own experience: as I was taught one way, that makes it the ‘way’ to do it. But let me try:

Think of the normal sized numbers as the ‘main’ number. Any sub- and superscripts indicate you have to do something to the main number. In 2^3, ‘2’ is the main number, The superscript after a number is the number of times you multiply the main number by itself.

Part of the confusion might be that computers suck at ‘typing’ math. I learned pre-computers and so I always saw 2 with a superscript 3 after it: either in textbooks or on the blackboard. “2^3” came into being when computers were new and the typefonts cou’dn’t show super- or subscripts.

Does any of this help?

@mrItty, This is my understanding of the terms of multiplication

<shrug> Your understanding, along with that document, are clearly the wrong way of thinking of multiplication, as you yourself proved, because that is not in alignment with the rest of mathematics (specifically, exponentiation). If you think about it the other way around, like I do, there is no confusion.

I have seen this argued both ways. To me, it is more aligned to the way we speak to have the multiplier first. x + x + x = 3x, not x3. We speak of 3 dozen eggs and 3 times as many people. The number giving the number of copies comes first.

If I asked you to do 123 * 127 longhand, you would interpret that as:
127
x 123
————-

The 127 is acted upon by the 123.

When speaking in words, sure, I agree. When writing it in symbolic mathematical form, I still say my way makes more sense. :-)

Consider then, the historical perspective. Mathematical notation was developed hundreds of years ago to express complex information in small spaces. Writing surfaces and writing instruments were costly to acquire. Consider, also, that mathematical notation was developed by adults for adults and was likely arrived at through trial and error or by consensus of a few philosophizing on the subject and shared from that point in time. These mathematical philosophers communicated with each other in person or in limited amounts of writing.

Then, also consider that your link shows math problems that are couched in terms to convey teaching concepts and in terms that beginners in logic need to translate their basic understandings into more complex understandings.

Mathematical notation is confusing and frustrating to the uninitiated and is also used by teachers to determine who has a ‘gift’ for math and who needs to learn methods for ordering their mind. I wish I had had more teachers who grasped that their job was to work with the latter, those with the gift didn’t need more help, I did. Oh well.

Yes, somewhat. Sometimes I like extra parentheses even when they are redundant.