Use properties of logarithm and exponential to solve the following equations:

Question

a) 2+3^x−1 =18

b) ln(x² −4)=8

c) log₃(x−2)+log₃(x+1)=6

Answer 1

a) 2+3^(x-1)=18

 

3^(x-1)=16

log3^(x-1)=log16

(x-1)log3=log16

x-1=log16/log3

x-1=1.20411998/0.47712125

x-1=2.523719

x=3.523719

 

b) ln(x²-4)=8

 

x^2-4=e^8

x^2-4=2980.9579

x^2=2980.9579+4

x^2=2984.9579

x=√2984.9579

x=54.634768

 

c) log(3)(x-2)+log(3)(x+1)=6

 

log(3)([x-2][x+1])=6

(x-2)(x+1)=3^6

x^2-x-2=729

x^2-x-731=0

[1±√(1+2924)]/2

(1±54.08327)/2

55.08327/2 and -53.08327/2

x=190.99744

x=-26.541635

Answer 2

(a)

 

2 + 3^x-1 = 18

 

3^x-1 = 16

 

(x - 1)log3 = log16

 

x-1 = log16 / log3

 

x = 1 + log16 / log3

 

(b)

 

ln(x² - 4) = 8

 

x² - 4 = e⁸

 

x² = 4 + e⁸

 

x = √(4 + e⁸)

 

(c)

 

log₃(x -2) + log₃(x + 1) = 6

 

log₃[(x - 2) (x + 1)] = 6

 

(x - 2) (x + 1) = 3⁶

 

You should be able to finish from here.