Common Logarithms
The common logarithm (written: log x) of a number is the power to
which 10 must be raised to equal that number. The logarithm of 1000
(written log 1000) is 3 because 10 cubed (10^{3}) is equal to 1000.
1000=10 x 10 x 10 = 10^{3}
log 1000 = 3
Logarithms for numbers are generally found using a calculator since
the relationship between a number and its logarithm is not generally
obvious. Logarithms of numbers larger than one are positive while
logarithms of numbers smaller than 1 are negative. (Logarithms are
not defined for numbers less than zero.) Writing numbers in
scientific notation clarifies this point as the logarithm of a number
will be close in value to the value of its exponent.
log 3021 = 3.480
3021 = 3.021 x 10^{3}
log 3.021 x 10^{3} = 3.480
For numbers larger than one, the first digit of the logarithm is
equal to the exponent (when the number is written with one digit to
the left of the decimal point). For numbers smaller than one, the
first digit may be one less than the value of the
exponent.
log 0.0000378 = 4.132
0.0000378 = 7.38 x 10^{5}
log 7.38 x 10^{5} = 4.132
Antilogarithms
If the logarithm is known, the corresponding number is found by
raising 10 to the logarithm (antilogarithm). Suppose the
logarithm of an unknown number is 3.63. The unknown number is found
as follows:
log x = 3.63
10^{log x} = 10^{3.63}
x = 4.27 x 10^{3}
Natural Logarithms
Natural logarithms (written: ln x) are identical to common
logarithms, except the natural log of a number is the power to which e
(2.718...) must be raised to get that number. While this number
appears arbitrary it appears in many mathematical formulas.
Natural logarithms and common logarithms are related as follows:
ln x = ln 10 x log x = 2.303 log x
The natural antilogarithm of x is found by raising e to the power
of x. Raising e to a power is called taking the exponential.
Logarithmic and Exponential Relationships
Some useful properties of logarithms:
Some useful properties of exponents:
Try some problems using logarithms!
Do online problems!
Manipulating Formulas Containing Logarithms
The general rules for manipulating formulas apply to formulas
containing logarithms. At times, these formulas may seem more
confusing. The appropriate application of logarithms and
antilogarithms are used to isolate the variable of interest.
Example
Rearrange the following equation to find E when k, A, T and R
are known.
 Start with the original formula:
 Examine the formula. What changes need to be made to obtain an
expression for E in terms of the other variables?
 Determine the final formula:
Example
Rearrange the following equation to find A when k, E, T and R
are known.
 Start with the original formula:
 Examine the formula. What changes need to be made to obtain an
expression for A in terms of the other variables?
 Add E/(RT) from both sides:

Now the expression is for ln A. To remove the natural log take
the exponential of each side:
 Determine the final formula:
The formula may be further simplified using the properties of
exponents given above:
Try some problems manipulating formulas containing logarithms!
Do online problems!
