Simplifying Logarithms: Troubleshooting Common Mistakes

[Rectifier]

Hey there!
I am getting two completely different equations when I try to simplify one.
What am I doing wrong?

1.
$$y=ln(2x) \ \Leftrightarrow \ y=ln(2) + ln(x) \ \Leftrightarrow \ e^y=e^{ln(2)}+e^{ln(x)} \ \Leftrightarrow \ e^y=2+x$$

2.
$$y=ln(2x) \ \Leftrightarrow \ e^y=e^{ln(2x)} \ \Leftrightarrow \ e^y=2x$$

I am sorry if its something completely obvious. Its pretty late here so my brain doesn't function properly :)

 

[Mark44]

Hey there!
I am getting two completely different equations when I try to simplify one.
What am I doing wrong?

1.
$$y=ln(2x) \ \Leftrightarrow \ y=ln(2) + ln(x) \ \Leftrightarrow \ e^y=e^{ln(2)}+e^{ln(x)} \ \Leftrightarrow \ e^y=2+x$$

It should be $e^{ln(2) + ln(x)} = e^{ln(2)} \cdot e^{ln(x)}$ in your third step.

2.
$$y=ln(2x) \ \Leftrightarrow \ e^y=e^{ln(2x)} \ \Leftrightarrow \ e^y=2x$$

I am sorry if its something completely obvious. Its pretty late here so my brain doesn't function properly :)

 

[Rectifier]

Oh! Thank you Mark!