solving logarithmic equations?

Question

Can someone help me write this as a single log?

3[ ln x - 2 ln (x^2 + 1) ] + 2 ln 5

Answer 1

sure! :)

3lnx - 6ln(x^2+1) + 2ln5
ln(x^3) - ln[ (x^2+1)^6 ] + ln(5^2)
ln [ 25x^3 / (x^2+1)^6 ]

:)

Answer 2

First, get rid of the square bracket:

3[ ln x - 2 ln (x^2 + 1) ] + 2 ln 5
= 3 ln x - 6 ln (x^2 + 1) + 2 ln 5

Using the logarithmic rules, multiply 6 ln (x^2 + 1) + 2 ln 5:

6 ln (x^2 + 1) + 2 ln 5
= ln 25 (x^2 + 1)^6

Now, the problem looks something like:

3 ln x - ln 25(x^2 + 1)^6

Using the logarithmic rules again, make it into a single ln:

3 ln x - ln 25(x^2 + 1)^6
= ln x^3 / 25(x^2 + 1)^6

Answer 3

3[ln(x) - 2ln(x^2 + 1)] + 2 ln(5)

First, distribute the 3.

3ln(x) - 6ln(x^2 + 1) + 2ln(5)

Move the 3, the 6, and the 2 inside the logs.

ln(x^3) - 6ln( [x^2 + 1]^6 ) + ln(5^2)

ln(x^3) - 6ln( [x^2 + 1]^6 ) + ln(25)

First, combine the first two logs as a quotient.

ln ( x^3 / (x^2 + 1)^6 ) + ln(25)

Now, combine these two logs as a product.

ln( 25x^3 / (x^2 + 1)^6 )

Answer 4

= ln [x / (x² + 1)² ] ³ + ln 25
= ln 25 [ x / (x² + 1)² ] ³
= ln [ 25.x³ / (x² + 1)^6 ]