Solve the following equations e3x5 7 log8x 5 log8x 2 1

Solve the following equations. e^3x/5 = 7 log_8(x + 5) - log_8(x - 2) = 1 x^2/3 - 5x1/3 -6 = 0

Solution

a)e^3x/5 =7

by properties of logarithams a^b=c =>b =log_ac

3x/5 =ln7

3x =5(ln7)

x =(5/3)(ln7)

b)log₈(x+5)-log₈(x-2)=1

by properties of logarithams log x -log y =log x/y

log₈(x+5)/(x-2)=1

(x+5)/(x-2)=8¹

(x+5)/(x-2)=8

(x+5)=8(x-2)

x+5=8x-16

8x-x =5+16

7x=21

x=3

c)x^2/3-5x^1/3-6=0

let x^1/3= t =>x^2/3=t²

t²-5t -6 =0

(t-6)(t+1)=0

t +1=0 , t-6=0

t =-1 ,t =6

x^1/3=-1,x^1/3=6

x =(-1)³,x=6³

x =-1, x =216