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QUESTION

# How do you condense 20lna-4lnb?

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

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.