How do you condense #1/ 7 ln(x + 2)^7 + 1/ 2 ln x − ln(x^2 + 3x + 2)^2#?

1 Answer
Jim G.
Sep 14, 2016

#ln((x^(1/2)(x+2))/(x^2+3x+2)^2)#

Explanation:

Using the #color(blue)"laws of logarithms"#

#color(orange)"Reminder " color(red)(bar(ul(|color(white)(a/a)color(black)(logx^nhArrnlogx)color(white)(a/a)|)))#

#rArr1/7ln(x+2)^7=ln(x+2)^(7xx1/7)=ln(x+2)#

and #1/2lnx=lnx^(1/2)#

#color(orange)"Reminder " color(red)(bar(ul(|color(white)(a/a)color(black)(logx+logy=log(xy))color(white)(a/a)|)))#

#rArrln(x+2)+lnx^(1/2)=lnx^(1/2)(x+2)#

#color(orange)"Reminder " color(red)(bar(ul(|color(white)(a/a)color(black)(logx-logy=log(x/y))color(white)(a/a)|)))#

#rArrlnx^(1/2)(x+2)-ln(x^2+3x+2)^2=ln((x^(1/2)(x+2))/(x^2+3x+2)^2)#

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