Question

How do you condense `20lna-4lnb`?

Answer 1

`ln(a^20/b^4)` or `4ln(a^5/b)`.

Explanation:

`20lna=lna^20` by the rule `bloga=loga^b`.

Similarly, `4lnb=lnb^4` by the same rule.

So now we have `20lna-4lnb=lna^20-lnb^4`.

Now, using the rule `loga-logb=log(a/b)`, we can rewrite this as `ln(a^20/b^4)`.

If you like, you can write this as `ln((a^5/b)^4)`, which can be rewritten as `4ln(a^5/b)`.

You can plug numbers in to these rules to make sure they work. For example, `log_2 32-log_2 8=5-3=2`. If you use the division rule, you get `log_2 32-log_2 8=log_2(32/8)=log_2(4)=2`.

More generally, we can show this rule and the others to be true. I will show that `loga+logb=log(a*b)`.

Start with `y=lna+lnb`.

Exponentiate both sides to get `e^y=e^(lna+lnb)`.

Through rules of exponents, the right side can be rewritten to get `e^y=e^lna*e^lnb`.

Since `e^lnx=x` (if you're confused why, say something) we get `e^y=a*b`.

To finish, take the natural log of both sides to get `y=ln(a*b)`.

Similar things can be done to prove the other rules true.