1. ## Properties of logarithms

Hey guys, we're starting a new lesson on properties of logs tomorrow and I'm wondering if you guys can help my understand it? I wanted to do some advanced reading and I picked 3 random examples from the textbook.

1) ln(x+4/x-2)-2ln(x-2)

2) 1+log312-(1/2)log318-log92

3) 8log(b^2)√x - logb (xy)

BTW, the symbols in smaller font are supposed to mean 'to the base of' so and so. As always, thanks for your help!

2. Hello, archistrategos214!

If you're just starting, these are awful problems!
I assume you know the basic properties of logarithms.

1) .ln[(x+4)/(x-2)] - 2·ln(x - 2)
We have: .ln(x + 4) - ln(x - 2) - 2·ln(x - 2) .= .ln(x + 4) - 3·ln(x-2)

. . = .ln(x + 4) - ln(x - 2)³ .= .ln[(x + 4)/(x - 2)³]

2) .1 + log312 - ½·log318 - log92
That "base 9" really messes up the problem!

Assuming you don't know the Base-Change Formula yet,
. . I'll show you a primitive approach.

Let .log
9(2) = p . . . Then: .9^p .= .2 . . (3²)^p .= .2 . . 3^{2p} .= .2

. . Hence: .2p .= .log
3(2) . . p .= .½·log3(2)

Therefore: .log
9(2) .= .½·log3(2)

The problem becomes: .log
3(3) + log3(12) - ½·log3(18) - ½·log3(2)

. . = .log
3(3·12) - ½[log3(18) + log3(2)] .= .log3(36) - ½[log3(18·2)]

. . = .log
3(36) - ½log3(36) .= .½·log3(36) .= .log3(36^½) .= .log3(6)

3. Thanks for the help, Soroban! Yes, we're only starting this lesson today, but hopefully our instructor won't be a sadist and give us questions that are too hard. Again, thanks!

4. Hello again, archistrategos214!

I'll try to explain #3 . . . another ugly problem!

3) .8·log(√x) - logb(xy)

Let log
(√x) .= .p . . (b²)^p .= .√x . . b^{2p} .= .√x

Then: .2p .= .log
b(x^½) . . 2p .= .½·logb(x) . . p .= .¼·logb(x)

Hence: .log
(√x) .= .¼·logb(x)

The problem becomes: .8·¼·log
b(x) - logb(xy) . = . 2·logb(x) - logb(xy)

. . = .log
b(x²) - logb(xy) .= .logb(x²/xy) .= .logb(x/y)