Write the expression as one logarithm lnx2y3 minus 2 lnx Squ

Write the expression as one logarithm. ln(x^2y^3) minus 2 ln(x Squarerooty) + 3 ln(x^2/3 Squarerooty). Solve the equation. log(x^2 + 4) minus log(x + 2) = 2 + log(x minus 2)

Solution

5) ln(x^2y^3) -2 ln(xsqrt(y) + 3ln(x^2/y^1/3)

Use the log properties: alnx = lnx^a

lna+lnb = ln(a*b) ;lna - lnb = ln(a/b)

ln(x^2y^3) -ln(x^2 y) +ln(x^6/y)

ln(x^2y^3/x^2y) + ln(x^6/y)

ln(y^2) +ln(x^6/y)

ln(x^6y^2/y)

ln(x^6*y)

6) log(x^2 +4) -log(x+2) = 2+ log(x-2)

log(x^2 +4)/(x+2) = 2+log(x-2)

log(x^2 +4)/(x+2) - log(x-2) = 2

log(x^2+4)/(x+2)(x-2) =2

log(x^2+4)/(x^2-4) =2

(x^2 +4)/(x^2 -4) = 10^2

(x^2 +4) = 100(x^2 -4)

x^2 +4 = 100x^2 -400

-99x^2 = -404

x^2 = 404/99

x = sqrt(404/99) = 2.02

x= 2.02