Base e, In, Log

Date: 03/08/2002 at 05:28:55
From: Jerry Harper
Subject: Base e, In, Log

First of all, I would like to thank you for telling us, how the value 
of base e works. 

I have some problems below, in base e. I've tried everything, but 
can't come up with the same answer as the book, and they don't give a 
clear example, or next to none of how they got their answer. Would 
you give me the complete clue on how they got their answer please? 

3e^(2x-1) = 7    the textbook answer is 1.079 
                  
I have tried:    Div^Log(7+3) 2Log3  = 1.047....

                 Div^(log7+log3) 2Log3 = 1.3856

                 Div^(Log(e)7+Log(e)3) 2Log(e)3 = 1.666...


e^(x+1) = 8      the textbook answer is .924 

I tried a similar approach, but no luck!

We very much appreciate your site!

Date: 03/08/2002 at 09:07:17
From: Doctor Peterson
Subject: Re: Base e, In, Log

Hi, Jerry.

Thanks for your comments! We're always happy to see people both 
reading our site and asking questions. 

You solve this kind of problem just the way you solve other basic 
algebra problems: undo what is done on the left, one step at a time, 
until you get the variable alone. Note what is being done:

    3 e^(2x-1) = 7

If you knew the value of x, you would

    1. multiply by 2
    2. subtract 1
    3. raise e to that power
    4. multiply by 3

and you'd get 7.

How can we undo this? Think about putting on and taking off your 
shoes: in the morning, you first put on your socks, then your shoes; 
in the evening you take off your shoes, then your socks. You undo each 
step in the reverse order. Same here; undo each step this way:

    4. divide by 3
    3. take the natural log
    2. add 1
    1. divide by 2

Here we go:

    3 e^(2x-1) = 7

    e^(2x-1) = 7/3

    2x-1 = ln(7/3)

    2x = ln(7/3) + 1

    x = [ln(7/3) + 1]/2

      = [ln(2.3...] + 1]/2

      = [0.8473 + 1]/2

      = 1.8473/2

      = 0.9236

I think you must have copied the wrong answers!

Of course, you can (and always should) check your answer by plugging 
it back in:

    3 e^(2*0.9236-1) = 3e^0.8473 = 3*2.333... = 7

See if you can do the other problem the same way.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/