Properties of Logarithms

Properties of Logarithms:

Change of Base Formula:

One of the dilemmas is that your calculator consists of only logarithms for two bases on it. Base 10 (log) and base e (ln). What is to occur if you wish to know the logarithm for certain other base? Are you out of luck?

No, there is a change of base formula for transforming between various bases. To determine the log base a, where a is most probably some number other than 10 or e, or else you would just utilize the calculator.

Take the log of argument divided by the log of base.

loga x = (logb x)/(logb a)

There is no requirement that either base 10 or base e be employed, but as those are the two you have on your calculator, those are most likely the two that you are going to utilize the most. The natural log is favored (ln is just 2 letters while log is 3, plus there is the extra advantage). The base which you use does not matter, only that you utilize similar base for both the numerator and denominator.

loga x = (log x)/(log a) = (ln x)/(ln a)

Illustration: log3 7 = (ln 7)/(ln 3)

Logarithms are Exponents

Keep in mind that logarithms are exponents; therefore the properties of exponents are the properties of logarithms.


What is the rule whenever you multiply two values with similar base altogether (x2 * x3)? The rule is that you keep the base and add up the exponents. Well, keep in mind that logarithms are exponents, and whenever you multiply, you are going to add up the logarithms.

The log of a product is the sum of logs.

loga xy = loga x + loga y


The rule if you divide two values with the similar base is to subtract exponents. Thus, the rule for division is to subtract logarithms.
The log of a quotient is the difference of logs.

loga (x/y) = loga x - loga y

Rising to a Power:

Whenever you increase a quantity to a power, then the rule is that you multiply the exponents altogether. In this situation, one of the exponents will be log, and the other exponent will be the power you are increasing the quantity to.

The exponent on the argument is the coefficient of log.

loga xr = r * loga x

Melodic Mathematics:

Some of the statements above are much melodious. That is, they sound excellent. It might help you to remember the melodic mathematics, instead of the formula.

a) Log of a product is the sum of logs.

b) Sum of the logs is the log of products.

c) Log of a quotient is the difference of logs.

d) Difference of the logs is a log of quotient.

e) The exponent on argument is the coefficient of log.

f) The coefficient of log is the exponent on argument.

Common Mistakes:

a) The log of a sum is not the sum of logs. The sum of logs is the log of product. The log of a sum can’t be simplified.

loga (x + y) ≠ loga x + loga y

b) The log of a difference is not the difference of logs. The difference of logs is the log of quotient. The log of a difference can’t be simplified.

loga (x - y) ≠ loga x - loga y

c) The exponent on the log is not the coefficient of log. Only if the argument is increased to a power can the exponent be turned to the coefficient. Whenever the whole logarithm is increased to a power, then it cannot be simplified.

(loga x)r ≠ r * loga x

d) The log of quotient is not the quotient of logs. The quotient of logs is from the change of base formula. Log of a quotient is the difference of logs.

loga (x/y) ≠ ( loga x )/( loga y )

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