Is a log base two always going to be smaller than a log base 3?

Question

Is it always, sometimes, or never true that

log base 2 of x < log base 3 of x

for any value of x

Answer 1

Using change of base law:

log x = log x/log 2

2

x

log = log x/log 3

3

Since log 3 is greater than log 2, log base 2 of x will never be less than log base 3 of x.

Answer 2

Test this.

p = log₂x ⇒ x = 2^p

q = log₃x ⇒ x = 3^q

2^p = 3^q

Take log on both sides to any base, as long as it is the same base on both sides.

p log2 = q log3

p/q = log3 / log2 > 1

p > q