# Log simplification question

• UniPhysics90
In summary, the conversation discusses a worked example involving using Stirlings approximations. The initial log equation is simplified to -kNcln(c), with the help of a factor of N inside the last logarithm and the fact that c is small. The conversation also suggests that neglecting the ln(N) terms relative to -ln(c) is a reasonable assumption, with N being much smaller than 1/c.

#### UniPhysics90

This is from a worked example involving using Stirlings approximations.

I have the log equation, where c<<1:

S=K(Nln(N)-cNln(cN)-N(1-c)ln(1-c))

In the next line of the example, this is simplified to

S=-kNcln(c)

I've tried a few ways of getting to this, but have had no success. I've tried making the last term 0, but still don't get the right answer.

Any help would be great :) Thanks

First off, I think you're missing a factor of N inside the last logarithm. I.e., your expression (aside from the Boltzmann constant) is

$$N\ln N - cN\ln(cN) - N(1-c)\ln(N(1-c))$$

From there a bunch of stuff should cancel, leaving you with

$$-Nc\ln c - N(1-c)\ln(1-c)$$.

Using the fact that c is small, you can show that the first term is much larger than the other term, hence you can neglect the other term.

It looks like they are using

ln(cN) = ln(c) + ln(N)​

and then neglecting the ln(N) terms relative to -ln(c), i.e. N « 1/c.

UniPhysics90, from the context of what you are reading, does that seem a reasonable assumption?

## What is log simplification?

Log simplification is the process of rewriting a logarithmic expression in a simpler form. This is done by using properties of logarithms such as the power rule, product rule, quotient rule, and change of base formula.

## Why is log simplification important?

Log simplification is important because it allows us to solve equations and problems involving logarithms more easily. It also helps to make complex expressions more manageable and easier to work with.

## What are some common properties used in log simplification?

The most common properties used in log simplification are the power rule, product rule, quotient rule, and change of base formula. These properties allow us to rewrite logarithmic expressions in a simpler form.

## What are some tips for simplifying logarithmic expressions?

Some tips for simplifying logarithmic expressions include factoring out common factors, using properties of exponents to rewrite the expression, and combining logarithmic terms using the rules of logarithms.

## Are there any common mistakes to avoid when simplifying logarithmic expressions?

One common mistake to avoid when simplifying logarithmic expressions is incorrectly applying the properties of logarithms. It is important to remember the correct formulas and rules for simplifying logarithms. Another mistake to avoid is dividing by zero, as this is undefined in logarithmic expressions.