On logarithms and funny notations

I remember having trouble with logarithms in high school because I had to do a transformation that hurt my head to understand. Before reviewing this section of pre-calculus, I consulted my memory bank's folio on logs. It read: "Logs will ask you to do stuff to convert them that will feel unnatural." I could not remember what it was exactly, but I could see the numbers bouncing around each other rearranging themselves. As I reviewed the chapter I remember what that is.

 

Logarithms are just the accepted notation to answer this question: to what power do I need to raise x number to give me answer y? In other words, I might want to know to what power should I raise 2 to get 64. In exponential form I would write 2^x=64.

 

However, in logarithmic form we would grab the x exponent throw it across the equal sign, then take the 2 shrink it, append it to the word "log" and finally take the 64 across the equal sign to join log₂. At the end we should have something like this: log₂64=x.

I hope you can see why it blew my mind as a teenager.

 

The Log transformation got me to think about math notation, or the way we represent mathematical concepts. I mean, if you think about it, someone at some point had to decide that the letter x would make a good place holder for a number we don't know; or that a cross would signify addition and a a horizontal line subtraction. I bet those first proponents had to explain what the symbols meant:

"Dear colleagues, whenever you see this notation of two parallel horizontal lines "=" it means that the terms on either side of "=" are equal"

I can almost hear the collective "Ahhh"s of fellow mathematicians reveling in not having to write the word "equal" anymore. I am of course being a little facetious. I am bet the introduction of the equal sign did not happen quite that way...but come on, I bet the person who came up with the radical sign (√) for square roots got a long of grief from the long division enthusiasts.

Getting back to logarithms, the logarithmic and exponential, as best as I can explain from what I understood, equations are crucial to describe rapid growth (or decline). Imagine you had a magic ball that would split into two magic balls every hour, then the new balls would also split every hour and so on. How many balls will you have in 24 hours? Pick a number in your mind and make it high. If you number is 16.7 million balls, you are right on the money.

The exponential or logarithmic graph has a segment rises or falls very quickly and another that gets progressively smaller but never touches the point it gets close two. That, if I remember correctly is an asymptote. A basic exponential graph has the x-axis as an asymptote, the logarithmic graph usually has the y-axis.

If you look at the graphs of log₂(x)and 2^x , the first is logarithmic and the second exponential, they will be mirror images of each other along the y=x line. This happens because in the logarithmic equation, the y values you get are the power you need to raise 2 in order to get the x values, while in the exponential one the y values you get are the result of raising 2 to the x power.

The cousera course videos keep telling me that exponential and logarithmic equations are important to calculus. For the time being, I guess I must take their word for it.

 

If anyone thinks I am missing something, let me know.

 

'Till next time.