The Difference Between log_a b^n and [log_a b]^n

Hi guys, now, I will discuss about the mistake which often the students do in the concept of LOGARITHM.

According the survey which my lecturer doing, he finding that the students often assume that, 

 log_a b^n = [log_a b]^n

Actually, it happens cause students does not know that the logarithm has the specific properties which can not solved like ordinary multiplication operation. Now I will show you about the difference between ...

log_a b^n and [log_a b]^n ;

with a > 0, a ≠ 1, b > 0; a, b ∈ R

Before this, remember about the basic concept of multiplication, Multiplication is defined as meaning that you have a certain number of groups of the same size. Then, it can be solved by repeated addition.

n × m = m + m + ⋯ + m,

"m" in "n" times

So,

log_a b^n

= n ∙ log_a b 

= log_a b + log_a b  + ... + log_a b ,

"log_a b" in "n" times

How about [log_a b]^n ?

Do you still remember? The exponent of a number says how many times to use the number in a multiplication. You can multiply any number by itself as many times as you want using exponents.

In 9² the "2" says to use 9 twice in a multiplication,

so 9² = 9 × 9 = 72

In more example:

5³ = 5 × 5 × 5 = 125

In words: 5³ could be called "5 to the third power", "5 to the power 3" or simply "5 cubed"

So,

[log_a b]^n

= [log_a b]^n x [log_a b]^n x ... x [log_a b]^n,

[log_a b]^n in "n" factor.

So, we can get conclusion that:

log_a b^n ≠ [log_a b]^n