I don't know how to solve the following problems:
1.) ln(x^2 + 4) - ln (x + 2) = 2 + ln(x-2)
2.) log(57x) = 2 + log(x-2)
3.) log base 5 (x-5) = 5
Your assistance would be much appreciated.
i'll do the first, the others are similar, so try them yourself.
some rules to know:
rule 1: $\displaystyle \log_a b = c \Longleftrightarrow a^c = b$
rule 2: $\displaystyle \log_a x + \log_a y = \log_a xy$
rule 3: $\displaystyle \log_a x - \log_a y = \log_a \frac xy$
important fact: $\displaystyle \ln x \equiv \log_e x$
now:
$\displaystyle \ln (x^2 + 4) - \ln (x + 2) = 2 + \ln (x - 2)$
$\displaystyle \Rightarrow \ln \left( \frac {x^2 + 4}{x + 2} \right) = 2 + \ln (x - 2)$ ....................rule 3
$\displaystyle \Rightarrow \ln \left( \frac {x^2 + 4}{x + 2} \right) - \ln ( x - 2) = 2$
$\displaystyle \Rightarrow \ln \left( \frac {x^2 + 4}{(x + 2)(x - 2)} \right) = 2$ ........................rule 3 again
$\displaystyle \Rightarrow e^2 = \frac {x^2 + 4}{(x + 2)(x - 2)}$ ....................rule 1
now solve for $\displaystyle x$.
of course, with logarithms we have a lot of flexibility. there are usually several ways to do the problem. play around with the rules until you get used to them, and you will start to see these manipulations quite easily. incidentally, there are about 10 rules you need to know to make your life with logarithms easier. around 5 are compulsory and the other 5 are icing on the cake. you can do a search for them. try searching for "laws of logarithms" or something to that effect. they have been posted several times on this site