In Chemistry, we work with two types of logarithms. We encounter common (base 10) logarithms in General Chemistry, primarily when dealing with pH. An understanding of natural (base e) logarithms is necessary in some higher-level courses. Here, we will only discuss common logarithms.
The basic relationship is:
If: X = 10Y then log10(X) = Y
1. You can’t take the logarithm of a negative number or of zero.
2. The logarithm of a positive number may be negative or zero.
3. Different books and Tables use different notations: log(X) without the subscript may mean either log10(X) or loge(X).
natural logarithm of a number is always 2.30258...... times the common
logarithm of that number. ln (10) =
If you are given that the common
is an integer or a negative integer, you know that the number is 10
raised to that power:
If: log10(X) = -3,
X = 10-3
If you are given log10(X) = 1.9156, you know that
X = 101.9156 ,
and is between 101 (10) and 102
(100) - probably closer to 100. You can also say that it is a number between
1 and 10 multiplied by 101 (#.##...). If
you put 1.9156 into the right (Y) side of the calculator above, and
click on 10Y , the result is 82.3379..... , as we saw
If you are given log10(X) = -2.56, you know that
X = 10-2.56 ,
and is between 10-2 (0.01) and 10-3
(0.001). You can also say that it is a
number between 1 and 10 multiplied by 10-3 (0.00###...).
The calculator will show you that X =
2.754 x 10-3 (2.754...E-3) or 0.002754.
Operations with Logarithms
Logarithms change mathematical operations:
Multiplication is replaced by Addition:
log(X*Y) = log(X) + log(Y) ,
Division is replaced by Subtraction:
log(X/Y) = log(X) - log(Y) .
Roots and Powers become:
log(Xn) = n log(X) , log(X1/n) = (1/n) log(X) .
|Discussion of pH|