Solve the following equations algebraically Find the exact a

Solve the following equations algebraically. Find the exact answers (i.e. NOT the decimal approximation) 2(4^3-2x) + 21 = 71

Solution

1)

a)2(4^3-2x) +21=71

2(4^3-2x) =71-21

2(4^3-2x) =50

(4^3-2x) =25

take logaritham on both sides

ln(4^3-2x) =ln25

ln a^b=b lna

(3-2x)ln(4) =ln25

(3-2x) =ln25/ln4

lna/lnb= ln_ba

(3-2x) =ln₄25

2x =3-ln₄25

x=(1/2)(3-ln₄25)

b)2e^2x-e^x=15

2(e^x)²-e^x-15=0

let e^x=p

2p²-p-15=0

2p²-6p+5p-15=0

2p(p-3) +5(p-3)=0

(2p+5)(p-3)=0

2p+5=0,p-3=0

p=-5/2 , p=3

e^x=-5/2 not possible

e^x=3

x=ln(3)

c)log₄x +log₄(x+3) =log₄(2x+6)

log_ab+log_ac =log_abc

log₄x*(x+3) =log₄(2(x+3))

log₄x*(x+3) -log₄(2(x+3)) =0

log a-log b=log a/b

log₄x*(x+3)/(2(x+3)) =0

log₄x/2=0

x/2=4⁰

x/2=1

x=2