Once, long ago, there was the Land of Numbers. All the numbers lived here and they were happy...
... mostly. Except for the exponents.
For a long time, the exponents were treated like second-class citizens of numberland. They couldn't go anywhere without their base numbers, and they were never allowed to get quite as big as their bases. This made them so angry that they eventually started a revolution. They rabbleroused and robblehoused and eventually ended up at the door of the King of Numberland himself.
"We're forced to live as slaves!" the Head Exponent cried.
"YEAH!" chimed the other exponents.
"We can't live like this!" the Head Exponent declared.
"YEAH!" cried the other exponents.
"WE DEMAND A CHANGE!" proclaimed the Head Exponent.
"YEAH, YEAH!" chorused the other exponents.
The King of Numberland could see that his kingdom was in peril. He needed to do something. He called the base numbers in, and told them, "I'm sorry, but you're going to have to switch places with the exponents." The base numbers weren't happy with this, but what could they do?
"Hey, wait a minute," interjected the Head Exponent. "We've spent all this time riding on the shoulder of our bases. We deserve a little more than to switch places."
Beside himself, the King cried, "Well, what do you want?"
"We want to step on them!" the exponents cried! And the King, in the interest of preserving the peace in Numberland, agreed. Now, the base numbers REALLY weren't happy with this, but they didn't dare disobey the King.
And so the base numbers are forced to spend their days not only smaller than the exponents, but also being tread upon by them, with the help of Logarithm, the Great Mediator.
logaB is the same as aB
We usually work in base-10, so the LOG button on a calculator actually stands for log10.
And because we use loge so much, it's shortened into ln. That's all ln is - loge.
This story was told to me by my pre-calc professor, and has seriously helped me with dealing with logs in calc. So I'm sharing it in the hopes that it helps someone else as well. :)