The property of logarithm to change its base and retain its equality is really amazing. To unveil this mystery we need to know how it works. Knowing this property makes us capable of solving hard problems in logarithmic equations.
Important properties to take note when dealing with logarithm,
with the same base of 10.
Worked Problem 1: 2013 MMC Elimination
Write as a single logarithm.
Using the change of base formula we will be able to work with this problem easily.
Observe that we can cancel same terms leaving
but this can be simplified further using reversed change of base formula,
Worked Problem 2:
Solve for x:
In solving logarithmic equation we need to make sure first that the bases of expressions involved are the same. The base of the first expression is 4 and the other one is 2. So we need to find ways to make those bases the same. Right! We use change of base formula.
We need to express to base 2 or to base 4. It’s up to your choice.
Mine is the second one.
Expressing to base 4,
Going back to original equation,
Worked Problem 3:
Solve for x:
Using change base formula,
By cross multiplication,
The equation is simplified as:
Squaring both sides,
Plugging back the values of a and b,
(Note! They are equal by comparison, not by rule of logarithm.)
Worked Problem 4:
Let’s start with
Using change of base,
This will happen the same in other terms. Applying change of base to all expressions we have,
Take note that the denominator is the same.
Using multiplication-addition law mentioned above in numerator,