# Ln vs log, when to use?

1. Nov 14, 2017

### Drake M

1. The problem statement, all variables and given/known data
I'm having a hard time differentiating when to use log instead of ln, vice versa. Are there any general rules to follow?

For example I have to evaluate 4u^-3 + u^-1.

2. Relevant equations
f'(1/u) = log u
f'(1/u) = ln u

3. The attempt at a solution
I put -2u^-2 + log(u) but the textbook solution shows the same answer except with ln.

2. Nov 14, 2017

### FactChecker

The derivative of 1/u is ln(u), not log10(u). When you are dealing with derivatives or integrals, the natural log has an advantage. In those situations, log10 requires that you correctly include factors of ln(10) in your answers.
CORRECTION: The derivative of ln(u) is 1/u. ( Thanks @lcgldr)

Last edited: Nov 16, 2017
3. Nov 14, 2017

### Ray Vickson

Some books and papers denote the natural logarithm of $u$ as $\log(u)$ instead of $\ln(u)$. Some other books reserve the notation $\log(u)$ for log to base 10 of $u$. Still others use $\log_{10}(u)$ instead in that context.

I do believe that the most common forms are $\ln(u)$ for $\log_e(u)$ and $\log(u)$ for $\log_{10}(u)$, but I have no statistics on that issue.

Some computer algebra systems adopt similar standards. For example, Maple accepts either "log(u)" or "ln(u)" for $\ln(u)$ but prints it out as $\ln(u)$. It needs the fancier notation "log[a](u)" for $\log_a(u)$ when $a \neq e$. I don't have access to Mathematica, so I don't know what conventions it adopts.

4. Nov 15, 2017

### FactChecker

Just for clarity, I recommend the use of ln() for base e and loga() for arbitrary base a. Otherwise, it leaves people guessing what base you want. It's nicer to specify the base when you use 'log'.

The exception is in math books like complex analysis, where the base is always e, regardless of the 'ln', versus 'log' choice.

5. Nov 15, 2017

### hilbert2

I think the notation $\ln u$ is common in applied, less rigorous mathematics, especially senior high school level math. Pure mathematics literature usually assumes that $\log u$ means the base $e$ logarithm unless otherwise stated. The base-10 logarithm is important in chemistry, where the pH scale is defined with it.

6. Nov 16, 2017

### rcgldr

7. Nov 16, 2017

### FactChecker

8. Nov 26, 2017

### Drake M

I meant to thank you all for your help sooner but i got caught up in test and mid-terms. Thanks!