Suppose that x is a positive real number. The common logarithm of x can be expressed in
terms of the natural logarithms of x and 10. If we go to six decimal places to approximate the
natural log of 10, we have
log 10 x= (ln x) / (ln 10)
≈ (ln x) / 2.302585
≈ 0.434294 ln x
Let’s try a numerical example. Let x= 3.537. Working to three decimal places, we can use a
calculator to find
ln 3.537 ≈ 1.263We multiply by 0.434294 and round the answer off to three decimal places, getting
0.434294× 1.263 ≈ 0.549
We can compare this with the common log of 3.537 as a calculator determines it, again
rounding off to three decimal places:
log 10 3.537 ≈ 0.549Now let’s go the other way. Suppose x is a positive real number. The natural logarithm of x can
be expressed in terms of the common logarithms of x and e. If we go to six decimal places to
approximate the common log of e, we have
ln x= (log 10 x) / (log 10 e)
≈ (log 10 x) / 0.434294
≈ 2.302588 log 10 x
If you’re astute, you’ll notice that this value differs slightly from the constant 2.302585 we
obtained above. Again, this is an example of rounding error. To demonstrate with a calculator,
the reciprocal of 0.434294 comes out as 2.302588 when you round it off to six decimal places.
But if you take the constant derived from ln 10 earlier, which is 2.302525 when rounded to
six decimal places, its reciprocal comes out as 0.434294.
Are you confused?
“Is there any way,” you ask, “to eliminate rounding errors in calculations?” The answer is, “Yes, sometimes;
but it can be a little tricky.” When working with irrational numbers, or any other numbers where the
values can be approximated but never exactly written down, you can carry out your calculations to many
more decimal places than necessary until the very end, and then—but only then—round the values off to
the number of places you want.
How Logarithms Work 485