Powerful Patterns 12: Logarithms in Base √(2)

Look at the list in the virtual chalkboard below. You should notice a pattern...

log_√(2)(2) = 2

log_√(2)(4) = 4

log_√(2)(8) = 6

log_√(2)(16) = 8

log_√(2)(32) = 10

log_√(2)(64) = 12

log_√(2)(128) = 14

log_√(2)(256) = 16

The logarithms of the powers of 2 in base √(2) are consecutive even numbers! Here's why:

log_√(2)(2^x) = 2x

Also, there are some math rules about exponents, radicals & logarithms to consider in this math trick. For example, raising to the power of ½ gives you the square root of the base. Furthermore:

log_a^b(x) = (1/b)log_a(x)

So:

log_√(2)(x) = 2log₂(x)

Since:

√(a) = a^(1/2) & 1/(b/c) = c/b

(If b is equal to 1, then c/b = c)

If the exponent is negative, then you get the reciprocal of the base's power!

x^-y = 1/(x^y)

So:

log_√(2)(1/2) = -2

log_√(2)(1/4) = -4

log_√(2)(1/8) = -6

log_√(2)(1/16) = -8

log_√(2)(1/32) = -10

log_√(2)(1/64) = -12

log_√(2)(1/128) = -14

log_√(2)(1/256) = -16

This list shows the reciprocals of every power of 2 in the 1st list! Notice how the final answers are negative instead of positive this time; plus, they're still consecutive even numbers!

Speaking of patterns, maybe you already seen some other numerical math patterns that other mathematicians already discovered & published! (Besides me or you; any open-minded person can be a great mathematician, like me!)

Back to Index Page Back to Math Trick Menu