Find exact solution to log equation

Question

Find the exact solution of the below equations and show work.

 

log5 = 1 + log(x+2)

 

 

 

 

 

3^x-1 = 27^3x

 

 

 

 

e^{x+1 =} 10^x

 

 

 

 

 

Thanks so much for the help!!!

Answer 1

log5=1+log(x+2)

log5-log(x+2)=1

log[(5/(x+2)]=1

assuming the base is 10

10¹=5/(x+2)

10(x+2)=5

10x+20=5

10x=-20+5

10x=-15

x=-1.5

 

3^(x-1)=27^(3x)

3^(x-1)=(3³)^(3x)

3^(x-1)=3^(9x)

x-1=9x

-1=9x-x

-1=8x

x=-1/8

 

e^(x+1)=10^x

e(e^x)=10^x

e=10^x/e^x

e=(10/e)^x

ln(e)=ln[(10/e)^x]

1=xln(10/e)

x=1/[ln(10/e)]

x=1/[ln(10)-ln(e)]

x=1/[ln(10)-1] (exact solution)

x=1/(2.3025851-1)

x=1/(1.3025851)

x=0.7677041