What is ln e? Need help solving these

Question

Hi, I have no idea what ln e means. I know ln stands for natural log. These are the equations I need help with : 

 

lne^2x+7 = -8 

 

and 

 

 

ln(1/e)^x2 = 3 

Answer 1

First look at log₁₀10. That equals one because 10¹ = 10

Now, ln is that same as log_e. So ln e = 1 because e¹ = e

ln e^2x+7 = -8

(2x+7) ln e = -8           laws of logs

2x + 7 = -8                 ln e = 1

2x = -15                      subtract 7 from both sides

x = -7.5                      divide both sides by 2

 

ln (1/e)^x² = 3

ln (e^-1)^x² = 3

ln e^(-x²) = 3

-x² = 3

x² = -3

Not possible. Check the equation.