Different ways that Log appears

Correct, log_e(x) is identical to ln(x) and is given the name "natural logarithm".

However, e^x is not the same as the natural logarithm, it is in fact it's inverse. Explicitly,

[tex]\log_e\left(e^x\right) = \ln\left(e^x\right) = x[/tex]

And,

[tex]e^{\ln(x)} = x[/tex]

Similarly,

[tex]\log_a\left(a^x\right) = x[/tex]

And,

[tex]a^{\log_a(x)} = x[/tex]

I hope this helps clear things up for you.

 

Hi!

[tex]log_e(x)[/tex] means the same as ln(x).

You would pronounce it "logarithm, base e".

I suspect that the examiner is worried that some people use "log" for natural logs (I prefer that), and some for log-base-10, so he's written "[tex]log_e(x)[/tex]" to remove any doubt.

Sorry, I don't follow this part of your query.