for any positive real number
Common mistakes to watch out for include
The general strategy is to consolidate the logarithms using these properties, and then to take both sides of the equation to the appropriate power in order to eliminate the logarithms if possible.
Find if .
By the definition of the logarithm,
Another way to view this solution is that we took to the power of the left side and got , and took to the power of the right side and got . More complicated logarithmic equations are often simplified by exponentiating both sides.
Other equations can be simplified using other properties of logarithms. One difficulty that arises is that eliminating logarithms and solving the resulting equation can introduce spurious solutions. These solutions violate the principle that the argument of the log function must always be positive. It is generally wise to check solutions by plugging them into the original equation and making sure that both sides are defined.
Find all such that
Simplify the given equation as follows:
This produces two potential solutions . But note that is not an actual solution, as is undefined. The only actual solution is .
The step in the solution above that was not reversible was the first one: although is positive if , and are not themselves positive.
More complicated logarithmic equations often involve more than one base. It can help to introduce unknowns to solve for the logarithms first. Another useful identity is , especially since can be chosen to be whatever simplifies the problem.
Suppose are positive real numbers such that
Show that or .
Rewrite this as
where the logs are all to the base . This simplifies to
Let and . Then , so
So or . In the first case, , so . In the second case, , so , so .
Equations involving exponents can often be simplified by taking logarithms:
Solve for if
Take of both sides:
and substituting turns this into , or . So or , which leads to or .
Logarithms can also make computation easier in certain practical situations. For example, logarithms with base give information about the number of decimal digits in a number.
which is bigger, or ? How many decimal digits do these two numbers have?
So is bigger. The number of decimal digits of is , so has digits and has digits.
Note that this is much easier than multiplying the numbers and comparing them directly. For applications that require large numbers (such as RSA encryption), this is very important.