Question

If you know logs well can you please explain something to me?
Here is the problem:
rewrite and simplify the log expression:
ln(5e^6)
I have the answer which is 6+ln5
my question is how? I am just trying to understand how that problem gives that answer? when I look at the log properties I though it would be the power property but when i tried to solve it as such it didn't produce the same answer. can someone help me be less confused?

Answer 1

\[\ln(5e^x)=\ln(5)+\ln(e^6)\] and \(\ln(e^6)=6\)

Answer 2

I don't understand why it would be the product property that is used and not the power property because it has a power in the problem.

Answer 3

there are two properties used here

Answer 4

one is
\[\log(AB)=\log(A)+\log(B)\] so
\[\ln(5e^6)=\ln(5)+\ln(e^6)\]

Answer 5

the second property is that the natural log, log base e, is written as \(\ln(x)\) rather than \(\log_e(x)\) so you have \(\log_e(e^x)\) which is pretty clearly 6
the natural log is the inverse of the exponential function, so it is always true that
\[\ln(e^x)=x\]

Answer 6

thank you so much! I couldn't figure out how the book was getting the answer

Answer 7

as inverse functions, you always have
\[\ln(e^x)=e^{\ln(x)}=x\]
yw

Answer 8

thank you!!!

Answer 9

yw again