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Let x and y be any real numbers:
log x = y
x = log inv (y) = 10^y
Example:
pH =13.22 = -log [H+]
log [H+] = -13.22
[H+] = inv log (-13.22) = 10^(-13.22)
[H+] = 6.0 x 10-14 M
FINDING ANTILOGARITHMS using a calculator (also called Inverse Logarithm)
Sometimes we know the logarithm (or ln) of a number and must work backwards to find the number itself. This is called finding the antilogarithm or inverse logarithm of the number. To do this using most simple scientific calculators,
- enter the number,
- press the inverse (inv) or shift button, then
- press the log (or ln) button. It might also be labeled the 10x (or ex) button.
- Example 5: log x = 4.203; so, x = inverse log of 4.203 = 15958.79147..... (too many significant figures)
There are three significant figures in the mantissa of the log, so the number has 3 significant figures. The answer to the correct number of significant figures is 1.60 x 104. - Example 6: log x = -15.3;
so, x = inv log (-15.3) = 5.011872336... x 10-16 = 5 x 10-16 (1 significant figure)
Natural logarithms work in the same way:
- Example 7: ln x = 2.56; so, x = inv ln (2.56) = 12.93581732... = 13 (2 sig. fig.)
Application to pH problems:
- Example 8: What is the concentration of the hydrogen ion concentration in an aqueous solution with pH = 13.22? pH = -log [H+] = 13.22
log [H+] = -13.22
[H+] = inv log (-13.22)
[H+] = 6.0 x 10-14 M (2 sig. fig.)
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It depends on what calculator you are using. Often, shift log is inverse log, or natural log, but it could have other meanings. Please restate the question and specify the model number of the calculator.
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