How do you "expand" and "condense" problems containing log???

Question

I can't figure these out. Can you help? We're supposed to show all work, but I don't even know where to begin.

a. Expand:
log base 2 (a / b^2)

b. Condense:
1/3 log (x^3 + 1) + 1/2 log (x^2 + 1)

Thanks in advance for any help you can offer.

Answer 1

Expand here just means to use the laws of logarithms to convert the log of some rational expression into a sum of logs of irreducible expressions. Thus:

log₂ (a/b²) = log₂ a - 2 log₂ b

Condense means to do just the opposite:

1/3 log (x³+1) + 1/2 log (x²+1) = log (∛(x³+1)√(x²+1))

Answer 2

Adding logs is the same as multiplying their arguments together.
Subtracting logs is the same as dividing their arguments together.
The coefficient of the log becomes the power of the argument.
(*They must have the same base)
This is just a few of the law of logs. More can be seen in my sources.

1) log base 2 (a / b^2)
log base 2 (a) - log base 2 (b^2)
log base 2 (a) - 2*log base 2 (b)

2) 1/3 log (x^3 + 1) + 1/2 log (x^2 + 1)
log ((x^3 + 1)^(1/3)) + log ((x^2 + 1)^(1/2))
log ((x^3 + 1)^(1/3)*(x^2 + 1)^(1/2))

Hope this helps!

Answer 3

The basic formulas for logs are (base can be anything as long as it's the same for all terms in a given formula):

log(a/b) = log(a) - log(b)
log(a*b) = log(a) + log(b)
log(a^n) = n*log(a)

So for part (a) above:
log(base2) (a/b^2) = log(base 2) (a) - log(base 2) (b^2)
= log(base 2) (a) - 2*log(base 2) (b)

And part (b) is:
1/3 log(x^3 + 1) + 1/2 log(x^2 + 1) = log(cbrt(x^3+1)) + log(sqrt(x^2+1)) = log(cbrt(x^3+1)sqrt(x^2+1))

Where cbrt means "cube root of" and sqrt means "square root of".

Answer 4

I've gotta' ask: why are you expanding and condensing a log anyway? Is that even possible?Lol! Sorry, brain tired or I would help.

Answer 5

its its a log a/b then you expand it and make it log a - log b....if its a division, thne you expand it and make it subtraction..and visversa

if its addition log a+b then it becomes log a x log b..you mulitply and visversa..

its kinda like exponents when you have an exponent on the top of a fraction and one on the bottom, you subtract them and when you have a two exponents multiplied you add them..

addition with multipication and subtraction with division