[Step 12: Reciprocal Relationship: Does swapping the base and the content result in a reciprocal?] Learning Logarithm Mechanics in 1 Minute: 20-part series, overwhelming knowledge acquired in segments
Hello, this is KoTa.
log_2 3 and log_3 2. These two, where the numbers have been flipped, have a very convenient relationship. It is the property that **"multiplying them results in 1."**
In other words, one is the **"reciprocal (the denominator and numerator flipped)"** of the other.
log_a b = 1 / log_b a
Why does this happen? You can understand it in an instant by using the "change of base formula" learned in Step 10. Let's rewrite "log_a b" using a new base "b".
The numerator is log_b b (= 1, remember!)
The denominator is log_b a
Combined, it is 1 / log_b a
For example, the calculation log_8 2. Thinking "to what power do I raise 8 to get 2?" is difficult, but if you flip it and think of it as 1 / log_2 8, the denominator is "3", so you can immediately see the answer is 1/3.
Key points for this session
Swapping the base and the antilogarithm results in a reciprocal (fraction)..
Formula: log_a b × log_b a = 1
When the "base is a larger number," using this reciprocal technique makes calculations incredibly easy!
What is the next step?
Step 13: Common Logarithm (1): The world of base 10 (the world of decimal system)
Up until now, various bases like "2" and "3" have appeared, but the logarithm most used in the real world is the "base 10" logarithm. Why 10? I will introduce the ultimate tool that combines the "decimal system" we use daily with log!
See you then!