The difference between log and ln

Question

$$\dfrac{1}{2}\ln(x+7)-(2 \ln x+3 \ln y)$$

Our professor let's us solve this but i do not understand how $\ln$ works. He says it has same properties with $\log$ but i still don't get it. What's the difference of the two?

Answer 1

The use of the "ln" abbreviation for natural logarithm is a bad thing because it makes people think that "log" is one thing and "ln" is another thing, and ask what's the difference between the two.

The base-$10$ logarithmic function is a logarithmic function.

The base-$2$ logarithmic function is a logarithmic function.

The base-$e$ logarithmic function is a logarithmic function.

The difference is which number is the base.

Mathematicians writing "$\log x$" usually mean $\log_e x$, also called $\ln x$.

Calculators use $\log x$ to mean $\log_{10} x$. This is also used in some of the sciences when doing numerical things.

The reason for the importance of base-$10$ logarithms was made obsolete by calculators. In the early '70s, calculators became widespread. Before then, many books had tables of base-$10$ logarithms in an appendix. Suppose you wanted the logarithm of $123$ The table gave you logarithms of numbers between $1$ and $10$, so you found $\log_{10}1.23= 0.089905\ldots$ and concluded that $\log_{10} 123 = 2.089905\ldots\;{}$. You added $2$ to move the decimal point over 2 places. That's why base 10 was used: to make that possible. If you wanted the square root of $7$, you found the logarithm of $7$, divided by $2$, then found the antilogarithm in the same table. If you wanted to divide $319450231$ by $2673019201$, you found logarithms of both in the table, subtracted, and then found the antilogarithm. And so on.

The important theoretical question to ask about "$\ln$" is why $e=2.71828182846\ldots$ is the "natural" base to use. (Has someone posted that question here?) (When I raise that question and try to answer it in a calculus class, some students ask "Do we HAVE to know this?? Will it be on the test?". Next time someone does that, I'm going to say "Who cares?".)

Answer 2

The common logarithm, is the logarithm base 10. It is the inverse of the exponential function $10^x$. In Calculus and Precalculus classes, it is usually denoted $\log$.

The natural logarithm, is the logarithm base $e$. It is the inverse of the exponential function $e^x$. In Calculus and Precalculus classes, it is often denoted $\ln$.

In general, if $a\gt 0$, $a\neq 1$, then the inverse of the function $a^x$ is the "logarithm base $a$", $\log_a(x)$.

The "guiding formula" is $$\log_a(b) = r\text{ if and only if }a^r = b.$$ From these, the properties of the logarithmic functions follow:

1. $\log_a(xy) = \log_a(x)+\log_a(y)$: logarithm of a product is the sum of the logarithms.

   Why? Say $\log_a(x) = r$ and $\log_a(y)=s$. That means that $a^r = x$ and $a^s=y$. Then $xy = a^ra^s = a^{r+s}$, so $\log_a(xy) = r+s = \log_a(x) + \log_a(y)$.

2. $\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$.

   Why? Again, say $\log_a(x) = r$ and $\log_a(y) = s$. Then $a^r = x$, $a^s = y$, so $\frac{x}{y} = \frac{a^r}{a^s} = a^{r-s}$, which means $\log_a\frac{x}{y}=r-s = \log_a(x)-\log_a(y)$.

3. $\log_a(x^t) = t\log_a(x)$$.

   Why? If $\log_a(x)=r$, so that $a^r = x$, then $x^t = (a^r)^t = a^{rt}$, so $\log_a(x^t) = rt = t\log_a(x)$.

4. $\log_a(a^r) = r$ and $a^{\log_a(x)} = x$. Because $\log_a(x)$ and $a^x$ are inverses of each other.

In particular, $\ln$, which is $\log_{e}$; and using $\log$ for $\log_{10}$, we have these properties: $$\begin{align*} \log(xy) &= \log(x)+\log(y) &\qquad \ln(xy) &=\ln(x) + \ln(y)\\ \log\left(\frac{x}{y}\right) &= \log(x) - \log(y) &\ln\left(\frac{x}{y}\right) &= \ln(x) - \ln(y)\\ \log(x^a) &= a\log(x) & \ln(x^a) &= a\ln(x)\\ \log(10^x) &= x & \ln(e^x) &= x\\ 10^{\log(x)} &= x & e^{\ln(x)} &= x \end{align*}$$

It also gives you a way to go back and forth between any logarithm and any other logarithm: if $a$ and $b$ are two bases, both positive, both different from one, what is the relation between $\log_a(x)$ and $\log_b(x)$?

If $\log_b(x)=r$, then $b^r = x$. So $$\log_a(x)= \log_a(b^r) = r\log_a(b) = \log_b(x)\log_a(b).$$ So we get that $$\log_b(x) = \frac{\log_a(x)}{\log_a(b)}.$$

As Henning points out below, while $\ln$ is not ambiguous (it always denotes logarithm base $e$), $\log$ is ambiguous and its exact meaning depends on context. In more advanced mathematics courses, it is usual to use it to mean the natural logarithm; in computer science it is very often used to denote logarithm base $2$. For some applications it does not matter (for example, when analyzing complexity, since two different logarithms are just scalar multiples of each other).