# Natural Logarithm

1. Aug 31, 2015

### funlord

are they equal?

2. Aug 31, 2015

### BvU

The first is hard to recognize, but I would take them both as the fourth power of the natural logarithm of (x+3).

The fact they use ln instead of log is decisive.

The first notation is to be avoided: there are already two notations for the base of a logarithm: $^e\log y$ and $\log_e y$ for $\ln y$ and this looks too much like a third notation for the same, which it is NOT.

3. Aug 31, 2015

### funlord

when i tried it on symbolab.com i subtract them both and i'd come up with answer of 0

does that mean they are equal?

4. Aug 31, 2015

### BvU

I don't acknowledge that kind of authority in a website. But yes, they are equal for the reason that they describe one and the same thing: the fourth power of the natural logarithm of (x+3).

5. Aug 31, 2015

### funlord

ok, thank you very much

6. Aug 31, 2015

### Staff: Mentor

Just for the record, I have never seen this notation -- $^e\log y$. By "never" I mean in the past 55+ years. That's not to say that someone hasn't used it somewhere, but if so, it's certainly not in common usage. The notation $\log_e y$ is rarely used, since $\ln y$ is defined to mean log, base e, of y.

If someone were to write $\log^4 (x + 3)$, I would interpret this to mean the same as $(\log(x + 3))^4$ following the usual shorthand as used in powers of trig functions. I would also interpret the log base to be 10, but in some contexts the implied log base could be e or possibly 2, in computer science textbooks.

7. Aug 31, 2015

### BvU

I now miss how you DO write $^4\log 16 = 2$ ? With the rarely used notation ?

Writing $^4\log 16$ is pretty common in Europe...

Ah, wait, of course $^e\log y$ is rarely used because $\ln y$ exists. In fact $\log y =\ ^e\log y\$ for a lot of decent people (mathematicians, for one) !