Question

What is the inverse of h(x)=12e^2x? Based on our answer, write an expression equivalent to x where h(x)=10

Answer 1

solve \[x=12e^{2y}\] for \(y\) in three steps
a) divide both sides by 12
b) write in equivalent logarithmic form
c) divide both sides by 2

Answer 2

\[a) \frac{ x }{ 12 }=e^(2y)\]

Answer 3

Im not sure how to do B

Answer 4

yup

Answer 5

oh ok

Answer 6

\[e^A=B\iff A=\ln(B)\]

Answer 7

replace \(A\) by \(2y\) and \(B\) by \(\frac{x}{12}\)

Answer 8

\[2y=\ln \frac{ x }{ 12 }\] Is tha correct?

Answer 9

yes

Answer 10

although i would write \[2y=\ln(\frac{x}{12})\] since the log is a function

Answer 11

Oh. Then it's: \[y= \ln \frac{ x }{ 6 }\] ?

Answer 12

oh no!!

Answer 13

hold on, lets go slow

Answer 14

For Part C, divide by 2, am I wrong? ;o

Answer 15

you see how you wrote \[2y=\ln \frac{ x }{ 12} \] and i said "i would write \[2y=\ln \left(\frac{ x }{ 12}\right) \]

Answer 16

Oh yes. I thought that didn't make any difference :#. I guess now I see the value in parenthesis

Answer 17

it is , in english, the log OF (x over 12)
not "the log of x" over 12

Answer 18

like for example if \(x=3\) then it would be \[\log(\frac{3}{12})=\log(\frac{1}{4})\]

Answer 19

half of \[\ln(\frac{x}{12})\] is \[\frac{1}{2}\ln(\frac{x}{12})\]

Answer 20

Oh damn. I made a critical error

Answer 21

or even \[\frac{\ln(\frac{x}{12})}{2}\]

Answer 22

clear right? just like if \[f(x)=x^2\] then \[\frac{1}{2}f(x)=\frac{x^2}{2}\] not \[\left(\frac{x}{2}\right)^2\]

Answer 23

yes, very clear

Answer 24

ok good, then we are done

Answer 25

Wait. If we do the second part of the question. We do \[10=\frac{ 1 }{ 2 } \ln (\frac{ x }{ 12 })\]?

Answer 26

no

Answer 27

\[h(x)=12e^{2x}\] so \[h(x)=10\] means \[12e^{2x}=10\]

Answer 28

\\[h^{-1}=\frac{1}{2}\ln(\frac{x}{12})\] so you solve \[h(x)=10\] by taking \[h^{-1}(10)\] that was the point of the problem , now that you have the inverse, you can solve

Answer 29

is plain englsh if \(h(x)=10\) then\[x=\frac{1}{2}\ln(\frac{10}{12})\]

Answer 30

OH. And now we just plug it in the calculator, right?

Answer 31

WAIT. IT SAYS EXPRESSION. Thanks for clarifiyng!