for 10 points please help?

Question

rewrite the expression:
ln 7 +2ln x + 5 ln (x^2+9)
as a single gogarithm ln A

Answer 1

addition of logarithms is multiplication of the parenthetical numbers...

The coefficients raise the parentheses to the powers..
i.e.
5 ln (x) = ln(x^5)

ln (7)(x^2)((x^2+9)^5)

Answer 2

10^2 * 10^3 = 10^5
log(10^2) = 2 (in base 10)
log(10^3) = 3
log(10^5) = 5

to replace a multiplication by a log operation, one adds the logs.

therefore, when logs are added, the base operation is a multiplication.
* is a multiplication sign

2 ln(x) = ln(x) + ln(x) = x * x = x^2

ln(7) + 2 ln(x) = ln(7*x^2)
5 ln(x^2+9) = ln[(x^2+9)^5] (same as the 2 ln(x) example)

ln(7x^2) + 5 ln(x^2+9) = ln[(7x^2)*(x^2+9)^5]

A = (7x^2) * (x^2+9)^5

Answer 3

using the properties of logarithms, ie. ln A + ln B = ln (AB) and A ln B = ln B^A

ln 7 + ln x^2 + ln ((x^2 + 9)^5)

ln (7*x^2*(x^2 +9)^5)

ln (7x^2(x^2 + 9)^5)