Logarithms "real life" usage for multiplication of 2 numbers In an effort to finally get to grips with logarithms (I had one related post today) I am looking for "real life" applications. Here is what I have found, and it astonishes me - they say that logarithms are great for multiplying two "big" numbers. Of course, for now let's pretend that we can't use calculator/computer to do the job. They suggest that multiplying 734 times 213 is much easier with log table, then, say, doing a straightforward multiplication on paper, or in the mind, even though the log usage doesn't give you a correct result. 
link to the article
Please, help me to understand why logs are more useful. Is it only because adding together smaller numbers "sounds" easier?
Also I will be truly grateful for your help on finding meaningful, mathematically correct real life applications of logarithms. 
Thank you very much!
 A: The practical real-life application of logarithms in doing arithmetic came before the days of hand-held electronic calculators (if you are young, let me explain that a hand-held calculator was your cell phone plus its calculator app, minus the ability to text, take pictures, access the internet, spend money on other apps, or -gulp- talk on the phone).
Engineers would carry a slide rule, which is a pair of sticks with logarithmic-spaced rulings marked on them.  You could use a slide rule to multiply, say, $743\times 213$ in less than a second, getting the answer to three-digit accuracy.
As to on-paper calculations, although the theoretical complexity of multiplication of $b$-bit numbers is the same as that of addition which is $O(b\log b)$, divide-and-conquer algorithms are needed to achieve this efficiency, and they are too complicated to be practical by hand.
The usual on-paper multiplication takes $O(b^2)$ work, whle looking up logs and adding them is again $O(b\log b)$ so there can be a practical advantage.  
A: Yes, you got it correctly: adding two small numbers is much easier and faster than multiplying two big ones. Obviously nowadays it's not very likely that you have at hand a log table instead of a calculator (e.g. within the smartphone) so the most important real life applications of logs are others.
A: 
Above is a table of logarithms - such as would have occurred in the back of most high school Algebra II books.
This is how I would have used it to compute $345 \times 0.0582$
Look down the "N" column for "34" and intersect that row with the column labelled $5$. You now know that $\log 3.45 = 0.5378$. Hence $$\log 345 = 2.5378$$ 
Similarly $\log 5.82= 0.7649$. In scientific notation, $0.0582 = 5.82 \times 10^{-2}$. For better or for worse, negative logarithms were avoided. Hence we wrote $$\log 0.0582 = 8.7649 - 10$$
No, really!. Adding the logarithms, we get
\begin{array}{rrrr}
     2.5378 \\
   + 8.7649 & - 10 \\
\hline
   11.3027 &- 10
\end{array}
Which simplifies to $$\log(345 \times 0.0582) = 1.3027$$
As most always happens, $3027$ is not in the table. We will need to do some linear interpolation.
We make the following table
\begin{array}{|c|c|}
\hline
   2.000 & .3010 \\
       x & .3027 \\
   2.010 & .3032 \\
\hline
\end{array}
Ignoring the decimal points and superfluous digits, we compute
\begin{align}
   \dfrac{x-00}{10-00} &= \dfrac{27-10}{32-10} \\
   \dfrac{x}{10} &= \dfrac{17}{22} \\
   x &= \dfrac{170}{22} \\
   x &= 8
\end{align}
Hence $$\log 2.008 = .3027$$
We conclude $$345 \times 0.0582 = 20.08$$
In fact, $345 \times 0.0582 = 20.079$.
