[Step 10: Change of Base Formula (1): The Savior When Bases Are Different] Understanding Logarithms in 1 Minute: 20-Part Series for Overwhelming Knowledge
Hello, this is KoTa.
When the bases are different, such as in "log_2 8 + log_4 16" where the bases are "2" and "4", the addition and subtraction rules we've used so far cannot be applied.
That is where **the "Change of Base Formula"** comes in. You can bring in any new number "c" you like and force the base to be rewritten.
log_a b = (log_c b) / (log_c a)
The way to remember it is very simple.
**The original base (a)** goes to the "bottom" of the denominator.
**The original antilogarithm (b)** goes to the "top" of the numerator.
For example, let's change the base of log_4 16 to "base 2".
The denominator is log_2 4 (= 2).
The numerator is log_2 16 (= 4).
The answer is 4 / 2 = 2
Just by aligning the bases, the answer that wasn't visible becomes clear.
Key points for this lesson
If the bases are different, decide on a **new base "c"** yourself and put it into a fraction form.
Arrange it so that "the bottom number (base) goes to the bottom of the fraction" and "the side number (antilogarithm) goes to the top of the fraction."
The trick is to choose a new base "c" that matches the other bases appearing in the problem!
What is the next step?
Step 11: Change of Base Formula (2): Mastering the Fraction Form
For those who think, "I understand the logic, but I'm not good at fraction calculations..." don't worry. Next time, I will introduce an exhilarating technique using this formula to cancel out "multiplication between logs" by simplifying them!
See you then!