How to Simplify Logarithims??

Question

How do I simplify

ln [square root[(x-1) / ((square root)(x+1))]]


Also how to simplify

ln [(x^x+1 / (1-x^3)) * (2^x / 7x)]

I think the answer below is going a little too deep... I just need to make it so I don't have logarithms of products, quotients or powers

Answer 1

ln [sqrt[(x-1) / ((sqrt)(x+1))]]

sqrt of x is the same as x^(1/2)
so
ln [sqrt[(x-1) / ((sqrt)(x+1))]]
ln [ [(x-1) / ((x+1)^(1/2))]^(1/2) ]
distribute the exponent to the top and bottom of this fraction
and:
ln [ [(x-1)^(1/2) / (x+1)^(1/4)] ]

If they want the expansion form then
ln (x-1)^(1/2) - ln (x+1)^(1/4)
1/2 ln (x-1) - 1/4 ln (x+1) <======answer


Expand this too!
ln [(x^x+1 / (1-x^3)) * (2^x / 7x)]
ln (x^x+1) - ln (1-x^3) + ln (2^x / 7x)
ln (x^x+1) - ln (1-x^3) + [ln (2^x) - ln 7x]
ln (x^x+1) - ln (1-x^3) + [x ln (2) - {ln 7+ln x}]

Basically anything with division you change to subtraction
and anything with Multiplication you change to Addition.
Exponents are brought to the front.

More examples:
Ln(3x) = Ln(3) + Ln(x) <=====Multiply changed to Add
Ln(4/y) = Ln(4) - Ln(y) <=====Divide changed to Subtract
Ln(5^x) = x Ln 5 <=====Exponent brought to front

Answer 2

ln [ √[ (x-1)/√(x+1) ] ] =
(1/2) ln [ (x-1)/√(x+1)] =
(1/2) ln (x-1) - ln [ √(x+1)] =
(1/2) ln (x-1) - (1/2) ln (x+1)

ln [(x^x+1 / (1-x^3)) * (2^x / 7x)] =
ln ((x^x+1 / (1-x^3)) + ln (2^x / 7x) =
ln ((x^x+1 / (1-x^3)) + ln (2^x) - ln (7x) =
I think you're missing a ) in the 1st half, so assuming you mean:
ln ((x^x+1) / (1-x^3)) + ln (2^x) - ln (7x) =
ln (x^x+1) - ln (1-x^3) + ln (2^x) - ln (7x) =
ln (x^x+1) - ln (1-x^3) + x ln 2 - ln 7 - ln x

Answer 3

What you need to keep in mind are a few of the basic properties of logarithms:

ln(x/y) = ln(x)-ln(y)
ln(x*y) = ln(x)+ln(y)
ln(x^2) = 2*ln(x)

Applying those identities ought to get you exactly what you need.

Answer 4

ln [(x^x+1 / (1-x^3)) * (2^x / 7x)]

ln (x^x+1) - [ ln(1-x^3) + ln(2^x / 7x)]

ln (x^x+1) - ln(1-x^3) - ln(2^x / 7x)

ln (x^x+1) - ln(1-x^3) - [ln(2^x) -ln(7x)]

ln (x^x+1) - ln(1-x^3) - ln(2^x) + ln(7x)


ln (x^x+1) - ln(1-x^3) - xln(2) + ln(7x)



ln [square root[(x-1) / ((square root)(x+1))]]

ln (square root)(x-1) - ln (square root)(x+1)

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

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

1/2 (ln [(x-1)/(x+1)])

Answer 5

ln(a * b) = ln(a) + ln(b)

ln(a / b) = ln(a) - ln(b)

ln(a ^ b) = ln(a) * b

Note that sqrt(x) is a special case of x^y

Answer 6

Differenciate w.r.t. x , you will get such an easy solution if you know differntiation