#R, 38 37 bytes

f=function(x)if(x>1)1+f(log(x))else 0

Thanks @user5957401 for the extra byte!

Test cases:

> f(0)
[1] 0
> f(1)
[1] 0
> f(2)
[1] 1
> f(3)
[1] 2
> f(4)
[1] 2
> f(3814279)
[1] 3
> f(3814280)
[1] 4

R, 38 37 bytes

f=function(x)if(x>1)1+f(log(x))else 0

Thanks @user5957401 for the extra byte!

Test cases:

> f(0)
[1] 0
> f(1)
[1] 0
> f(2)
[1] 1
> f(3)
[1] 2
> f(4)
[1] 2
> f(3814279)
[1] 3
> f(3814280)
[1] 4

#R, 38 37 bytes

f=function(x)if(x>1)1+f(log(x))else 0

Thanks @user5957401 for the extra byte!

Test cases:

> f(0)
[1] 0
> f(1)
[1] 0
> f(2)
[1] 1
> f(3)
[1] 2
> f(4)
[1] 2
> f(3814279)
[1] 3
> f(3814280)
[1] 4

#R, 38 37 bytes

f=function(x)if(x>1)1+f(log(x))else 0

Thanks @user5957401 for the extra byte!

Test cases:

> f(0)
[1] 0
> f(1)
[1] 0
> f(2)
[1] 1
> f(3)
[1] 2
> f(4)
[1] 2
> f(3814279)
[1] 3
> f(3814280)
[1] 4

#R, 38 bytes

f=function(x)ifelse(x>1,1+f(log(x)),0)

Test cases:

> f(0)
[1] 0
> f(1)
[1] 0
> f(2)
[1] 1
> f(3)
[1] 2
> f(4)
[1] 2
> f(3814279)
[1] 3
> f(3814280)
[1] 4

#R, 38 37 bytes

f=function(x)if(x>1)1+f(log(x))else 0

Thanks @user5957401 for the extra byte!

Test cases:

> f(0)
[1] 0
> f(1)
[1] 0
> f(2)
[1] 1
> f(3)
[1] 2
> f(4)
[1] 2
> f(3814279)
[1] 3
> f(3814280)
[1] 4