Question

How do I solve this equation for x?
6e^2x=7e^4x

Answer 1

\[7e^{4x}-6e^{2x}=0\]
you can start by factoring

Answer 2

I make it easier to see
\[7e^{2x+2x}-6e^{2x}=0 \\ 7e^{2x}e^{2x}-6e^{2x}=0\]
you should see both the first and second term on the left hand expression have common factor

Answer 3

I know the answer to the question is 0 but I just need to know how to get there.

Answer 4

do you know how to factor?

Answer 5

Yeah I do

Answer 6

can you factor what I asked you to factor

Answer 7

In this equation I am confused on how to factor though

Answer 8

do you see both terms have e^(2x)?

Answer 9

Yeah I see that

Answer 10

\[e^{2x}(7e^{2x}-6)=0\]
just factor it out then :p

Answer 11

set both factors equal to 0 and solve for x

Answer 12

\[e^{2x}=0 \text{ or } 7e^{2x}-6=0\]
hint e^(2x) is never 0 first equation cannot be solved
you just have to look at second equation now

Answer 13

\[e^{2x}=\frac{6}{7}\]
can you see if you can solve from here?

Answer 14

I cant. Maybe you could show me but explain it as you go

Answer 15

do you know that ln(e^x)=x?

Answer 16

just take ln( ) of both sides

Answer 17

So ln(e^2x)/ln(6/7)

Answer 18

Dont you just divide like so ^^^

Answer 19

you changed equal sign to division sign?

Answer 20

Oh so it is just ln(e^2x) = ln(6/7)

Answer 21

\[e^{2x}=\frac{6}{7} \\ \text{ equal sign does \not turn into division sign \it is an equal sign } \\ \ln(e^{2x})=\ln(\frac{6}{7})\]
good now use the fact that exp and ln are inverse functions of each other

Answer 22

that is I mean y=e^x and y=ln(x) are inverse of each other
that is
ln(e^x)=x
or
e^(ln(x))=x for x>0

Answer 23

or you can use power rule for log

Answer 24

that is ln(x^r)=rln(x)

Answer 25

you bring power of the inside down in front

Answer 26

Woah wait so then I just go 2xln=ln6/7 ??? I am confused

Answer 27

well you are missing the inside of that log

Answer 28

what happen to the e

Answer 29

\[2x \ln(e)=\ln(\frac{6}{7})\]

Answer 30

by the way ln(e)=1
so you really just have 2x on the left hand side

Answer 31

Oh gotcha

Answer 32

So then what??

Answer 33

solve for x

Answer 34

you only have one last step

Answer 35

divide by 2?

Answer 36

you know you would solve something like 2x=14
by dividing 2 on both sides

yes exactly same thing here
divide 2 on both sides

Answer 37

Alright and I get x= -.077 as the answer :)

Answer 38

that is approximation

Answer 39

\[2x=\ln(\frac{6}{7}) \\ x=\frac{1}{2} \ln(\frac{6}{7})\]

Answer 40

that is exact if you wanted to know :p

Answer 41

sweet can you help me with one more? This one will be a lot quicker.

Answer 42

ok i will try

Answer 43

10=1.5e^x

Answer 44

So I already attempted it and I just did
ln(10)/ln(1.5)

Answer 45

and I got approximately .934

Answer 46

one sec
multitasking

Answer 47

ok

Answer 48

\[10=1.5e^x \\ \frac{10}{1.5}=e^x\]

Answer 49

as before just take ln( ) of both sides

Answer 50

and ln(a/b) doesn't equal ln(a)/ln(b)

Answer 51

Okay so ln(10)=ln(1.5)

Answer 52

that isn't a true equation

Answer 53

e^x=a
means x=ln(a)
assuming of course a>0

Answer 54

your a is 10/1.5

Answer 55

Okay so ln(10/1.5)

Answer 56

yep

Answer 57

Then just put that into the calculator and solve??

Answer 58

\[a=b \cdot e^x \\ \text{ first step isolate } e^x \\ \text{ do this by dividing } b \text{ on both sides } \\ \frac{a}{b}=e^x \\ \text{ then take } \ln( ) \text{ of both sides } \\ \ln(\frac{a}{b})=\ln(e^x) \\ \ln(\frac{a}{b})=x \ln(e) \\ \text{ recall } \ln(e)=1 \\ \ln(\frac{a}{b})=x\]

well I would leave it like this
but yeah if you want an approximation

Answer 59

So I calculated and got x is approximately 1.897

Answer 60

yep seems great

Answer 61

sweet, thanks:)

Answer 62

np