Student "edition" found at {csi dot journalspace dot com}.

Maybe I shouldn't have started this blog now, not with everything that's been going on.

On the first day of the thirteenth week of classes, in my Mathematical Methods One session, I discussed the definition of natural logarithms, giving a brief history of how the number e was arrived at, having its roots in compound interest.

I also discussed the three main properties of logarithms.

First of these are that the logarithm of a product is equal to the sum of the logarithm of its factors.

Second is that the logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator.

Rewriting a fraction as a product with one factor having an exponent of negative one, we derive the third property, which is that the logarithm of a base raised to a certain exponent is equal to the exponent multiplied by the logarithm of the base raised to one.

After that I went into the identity properties.

I started with the fact that e raised to the natural logarithm of a number is equal to that number, same with the natural logarithm of e raised to a number.

If for a certain logarithm, the base is one, then the number we are taking the logarithm of must also be one, regardless of the result. If the base is zero, the same applies. The number must be zero.

But the logarithm of one has three possible values. It could be that the base is one and the result is any number not zero. There is also a chance that the base is negative one and the result is an even number. Lastly, it exists that if the base is not one or negative one, the result is zero.

If the logarithm of a number is equal to one, then the base is equal to the number. If the logarithm of a number is zero, then the number must be zero regardless of the base.

Then we had exercises again. The students are getting somewhat tired of these, despite the fact that it is for these that the subject was increased to six hours a week. So I had to make it by pairs again, even though I increased the number of items.

In my DIFEREQ class afterwards, we had exercises on the last three methods for solving differential equations. I made this into group work, dividing the class into three teams, that they had to present on the board at the end of the class.

Session 716 has a result of zero here. Class dismissed.