Limits of Logarithms

$Logarithms!$

$Logarithms$ are pretty cool; they were used by the ancients to turn pesky long multiplication and division problems into some simple addition and subtraction. How does this work?

Well, let's say some evil teacher asked us to find $1234\times 5678$, and of course, no calculator! Doing this by hand with normal multiplication would take a while, and is just downright tedious! So, let's make this a bit better:

We know that for some $a$ and $b$, $10^{a}$$=$$1234$, and $10^{b}$$=$$5678$ Then this means that $log1234$$=$$a$ and $log5678$$=$$b$ And we see that $1234\times 5678$=($10^{a}$)($10^{b}$)=$10^{a+b}$. Then we have $log$($10^{a}$$10^{b}$)$=$$log$($10^{a+b}$)$=$$a+b=log$($10^{a}$)$+log$($10^{b}$).

In other words, $log(1234)(5678)$$=log(1234)+log(5678)$.

$Yes!$ We've found that the logarithm of a product of two numbers is just the sum of the logarithms of the two numbers.

Now, we can simply look up $log1234$ and $log5678$ in our trusty logarithm tables, add the two values, and find the number-say $c$-such that $log(c)=log(1234)+log(5678)$.

$Yay!$ we've outsmarted our evil teacher!:)

Logarithms in the form $\log_n x = y$ means that $n$ raised to the power $y$ is equal to $x$.

The question $\log_n y = ?$ asks "to what number should $n$ be raised to obtain $y$?"

$\bullet$ $x$ can be non-integers. See rational exponents.

$\bullet$ $\log_{10} x$ can be simply written as $\log x$

$\bullet$ $\log_{e} x$ is the same as $\ln x$

Example: $\log_{2} 2048 = 11$

Example: $\ln{e^7} = 7$

Practice questions coming soon!