NO-FRILLS MATH FOR GENERAL CHEMISTRY

LOGARITHMS and ANTILOGARITHMS

Gary L. Bertrand, Professor Emeritus
Department of Chemistry
Missouri University of Science and Technology
Discussion of pH

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The subject of logarithms is best covered in an algebra class, and there are many good explanations on the web:

http://www.math.utah.edu/~pa/math/log.html

http://www.sosmath.com/algebra/logs/log1/log1.html

http://www.purplemath.com/modules/logs.htm

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

X       10Y Y = log10(X)
1,000,000 106 6
100,000 105 5
10,000 104 4
1,000 103 3
100 102 2
10 101 1
1 100 0
0.1 10-1 -1
0.01 10-2 -2
0.001 10-3 -3
0.0001 10-4 -4
0.00001 10-5 -5
0.000001 10-6 -6
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Notes:

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).

4. The natural logarithm of a number is always 2.30258...... times the common logarithm of that number.  ln (10) = 2.30258.....

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If you are given that the common logarithm is an integer or a negative integer, you know that the number is 10 raised to that power:
 
If:   log10(X) = 2,   X = 102   or 100 . 

If:   log10(X) = -3,   X = 10-3   or 1/1,000.

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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 earlier.

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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.

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Mathematical 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