Question

Simplify the following (these are examples to help me work out the actual problems).
ln1
lne^7
log(subscript3)(1/3)

Answer 1

first off it is always true that \(b^0=1\) and therefore \(\log_b(1)=0\)

Answer 2

another way to put this is since \(e^0=1\) we know \(\ln(1)=0\) since \(\ln(x)=\log_e(x)\)

Answer 3

since \(e^x\) and \(\ln(x)\) are inverse functions, the composite is always the identity
in other words it is always true that
\[e^{\ln(x)}=x\]and \[
ln(e^x)=x\]

Answer 4

since \(b^{-1}=\frac{1}{b}\) we know
\[\log_b(\frac{1}{b})=-1\]

Answer 5

I need a little further elabortaion on the last one, how would the variable still work in the equation? would it become negative 3 due to b^-1. Eh, if you could workout a different example for me, that would be great.