Question

quick log question

Answer 1

why is -2 become e and what happened to the ln

Answer 2

hey hey
or you familiar with
\[\ln(a)=y \implies e^{y}=a\]
where a is greater than 0.
or that f(x)=ln(x) and g(x)=e^x are inverse functions?

Answer 3

oh i didnt know that

Answer 4

those are called inverse rules?

Answer 5

\[a^{\log_a(x)}=x \text{ for all } x>0 , a \in (0,1) \cup (1,\infty)\]

\[\ln(x)=\log_e(x)\]

\[\log_e(2(2x-1))=-2 \\ e^{\log_e(2(2x-1))}=e^{-2}\]

Answer 6

since f=e^x and g=ln(x) are inverse then
\[e^{\log_e(2(2x-1))}=2(2x-1) \text{ when } 2(2x-1)>0\]

Answer 7

\[\log_e(2(2x-1))=-2 \\ e^{\log_e(2(2x-1))}=e^{-2} \\ 2(2x-1)=e^{-2}\]

Answer 8

other examples using:
\[a^{\log_a(x)}=x \text{ with the domain restrictions mentioned above }\]

\[e^{\ln(5)}=5 \\ e^{\ln(x^2)}=x^2 \\ 10^{\log_{10}(x^2)}=x^2 \\ \text{ note: } \log_{10}(x) \text{ is sometimes just written as } \log(x) \\ 5^{\log_5(x^2+1)}=x^2+1\]

Answer 9

you can also use the instead of f(g(x))=x we could use g(f(x))=x

\[f(x)=e^x \text{ and } g(x)=\ln(x)\]

so \[f(g(x))=f(\ln(x))=e^{\ln(x)}=x \\ \\ g(f(x))=g(e^x)=\ln(e^x)=x \ln(e)=x(1)=x \\ \text{ so examples using the second one since we already did examples using the } \\ \text{ first one } \\ \ln(e^5)=5 \\ \log_5(5^6)=6 \\ \log_x(x^{\sin(x)})=\sin(x) \text{ with } x \in (0,1 ) \cup (1,\infty)\]

Answer 10

thanks