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}
To make the calculation process "easier", we modify this table. All of
what follows is only done mentally. After a while, it becomes a very
natural process. We start by removing the decimal points.
\begin{array}{|c|c|}
\hline
2000 & 3010 \\
x & 3027 \\
2010 & 3032 \\
\hline
\end{array}
Technically, $x$ should be $1000x$. Ignore that. It's just a place mark and calling it $y$ or something else, will ultimately be a waste of mental time and effort.
Next, The numbers in the left column all start with the digits 20 and the numbers in the right column all start with the digits 30. If they don't change, then we can ignore them.
\begin{array}{|c|c|}
\hline
00 & 10 \\
x & 27 \\
10 & 32 \\
\hline
\end{array}
Again, all of this is being done in your head. The $x$ above is not the same $x$ that we started with. Ultimately, that won't matter matter. From the answer we get for this $x$, the value of the original $x$ will be completely obvious.
Now we compute the value of $x$ in the above table.
\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}
Where we rounded the value of $x$ off to the nearest integer.
Now we work backwards.
\begin{array}{rrrrrr}
\hline
00 & 10 &\cdots & 3010 &\cdots & .3010\\
8 & 27 &\cdots & 3027 &\cdots & .3027\\
10 & 32 &\cdots & 3032 &\cdots & .3032\\
\hline
\end{array}
Hence $$\log 2.008 = .3027$$
We conclude $$345 \times 0.0582 = 20.08$$
In fact, $345 \times 0.0582 = 20.079$.
