SOLUTION: https://imgur.com/a/nHNVzJD Find log(base z)w

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: https://imgur.com/a/nHNVzJD Find log(base z)w       Log On


   

Question 1153082: https://imgur.com/a/nHNVzJD Find log(base z)w
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
log%28x%2Cw%29=24
log%28y%2Cw%29=40
log%28xyz%2Cw%29=12
-----------------------------------
log%28x%2Cw%29=24
log%28w%29%2Flog%28x%29=24
log%28w%29=24log%28x%29
log%28w%29=log%28x%5E24%29
w=x%5E24.....eq.1a

log%28y%2Cw%29=40
log%28w%29%2Flog%28y%29=40
log%28w%29=40log%28y%29
log%28w%29=log%28y%5E40%29
w=y%5E40....eq.2a

log%28xyz%2Cw%29=12
log%28w%29%2Flog%28xyz%29=12
log%28w%29=12log%28xyz%29
log%28w%29=log%28%28xyz%29%5E12%29
w=%28xyz%29%5E12....eq.3a


from 1a and 2a we have

x%5E24=y%5E40
x=root%2824%2C%28y%5E40%29%29
x+=+2sqrt%286%29%2Asqrt%28y%5E40%29..............1b

from 1a and 3a we have
x%5E24=%28xyz%29%5E12
x=root%2824%2C%28xyz%29%5E12%29%29
x+=+2sqrt%286%29%2Ax%5E6%2Ay%5E6%2Az%5E6.........2b

from 2a and 3a we have

y%5E40=%28xyz%29%5E12
y=root%2840%2C%28xyz%29%5E12%29
y+=+2sqrt%2810%29%2Asqrt%28x%5E12%2Ay%5E12%2Az%5E12%29
y+=+2sqrt%2810%29%2Ax%5E6%2Ay%5E6%2Az%5E6...........3b

from 1b and 2b we have

2sqrt%286%29%2Asqrt%28y%5E40%29=+2sqrt%286%29%2Ax%5E6%2Ay%5E6%2Az%5E6

sqrt%28y%5E40%29=+x%5E6+%2Ay%5E6+%2Az%5E6

y%5E20=+x%5E6+%2Ay%5E6+%2Az%5E6

y=+root%2820%2C%28x%5E6+%2Ay%5E6+%2Az%5E6%29%29

y=2sqrt%285%29%2Asqrt%28x%5E6%2Ay%5E6%2Az%5E6%29.......1c


from 3b and 1c we have

2sqrt%2810%29%2Ax%5E6%2Ay%5E6%2Az%5E6=2sqrt%285%29%2Asqrt%28x%5E6%2Ay%5E6%2Az%5E6%29
sqrt%2810%29%2Ax%5E6%2A+y%5E6%2A+z%5E6=sqrt%285%29%2Asqrt%28x%5E6%2Ay%5E6%2Az%5E6%29
%28x%5E6%2A+y%5E6%2A+z%5E6%29%2Fsqrt%28x%5E6+y%5E6+z%5E6%29=sqrt%285%29%2Fsqrt%2810%29

%28x%5E6%2A+y%5E6%2A+z%5E6%29%2Fsqrt%28x%5E6+y%5E6+z%5E6%29=1%2Fsqrt%282%29.......square both sides
%28x%5E12%2A+y%5E12%2A+z%5E12%29%2F%28x%5E6+y%5E6+z%5E6%29=1%2F2
%28x%5E6%2A+y%5E6%2A+z%5E6%29=1%2F2...........2c

go to
y=2sqrt%285%29%2Asqrt%28x%5E6+y%5E6+z%5E6%29.......1c, use 2c
y=2sqrt%285%29%2Asqrt%281%2F2%29
y=2sqrt%285%29%2A1%2Fsqrt%282%29
y=sqrt%2810%29....................3c

go to
x+=+2sqrt%286%29%2Ax%5E6+%2Ay%5E6+%2Az%5E6.........2b, use 2c
x+=+2sqrt%286%29%2A%281%2F2%29
x+=+sqrt%286%29

go to
x+=+2sqrt%286%29%2Ax%5E6+%2Ay%5E6+%2Az%5E6.........2b, substitute x and+y

sqrt%286%29+=+2sqrt%286%29%2A%28sqrt%286%29%29%5E6+%2A%28sqrt%2810%29%29%5E6+%2Az%5E6

sqrt%286%29+=+432sqrt%286%29+%2A1000%2Az%5E6

z%5E6=sqrt%286%29%2F%28432sqrt%286%29+%2A1000%29

z%5E6=1%2F432000

z=root%286%2C%281%2F432000%29%29

z=sqrt%285%29%2F600


now find base xyz:
xyz=+sqrt%286%29%2A+sqrt%2810%29%2A%28sqrt%285%29%2F600%29
xyz=+sqrt%283%29%2F60

go to log%28xyz%2Cw%29=12......eq.3, substitute xyz
log%28sqrt%283%29%2F60%2Cw%29=12
log%28w%29%2Flog%28sqrt%283%29%2F60%29=12
log%28w%29=12log%28sqrt%283%29%2F60%29
log%28w%29=log%28%28sqrt%283%29%2F60%29%5E12%29.....if log same
w=%28sqrt%283%29%2F60%29%5E12.........3c


then log%28z%2Cw%29 will be
log%28z%2Cw%29=log%28%28sqrt%283%29%2F60%29%5E12%29%2Flog%28sqrt%285%29%2F600%29%29
log%28z%2Cw%29=%2812+log%281200%29%29%2Flog%2872000%29
log%28z%2Cw%29=7.6

Answer by ikleyn(52878) About Me  (Show Source):
You can put this solution on YOUR website!
.

You are given


    log%28x%2C%28w%29%29 = 24,  log%28y%2C%28w%29%29 = 40,  log%28xyz%2C%28w%29%29 = 12.    (1)


They want you find log%28z%2C%28w%29%29.


Solution


From (1), you have, using the change of base formula


    log%28w%2C+%28x%29%29 = 1%2F24;  log%28w%2C+%28y%29%29 = 1%2F40  and  log%28w%2C+%28xyz%29%29 = 1%2F12.


Transform the last equation


    log%28w%2C++%28xyz%29%29 = 1%2F12 = log%28w%2C+%28x%29%29 + log%28w%2C+%28y%29%29 + log%28w%2C+%28z%29%29 = 1%2F24 + 1%2F40 + log%28w%2C+%28z%29%29.


It implies


    log%28w%2C+%28z%29%29 = 1%2F12 - 1%2F24 - 1%2F40 = 10%2F120+-+5%2F120+-+3%2F120 = 2%2F120 = 1%2F60.


Hence,  log%28z%2C+%28w%29%29 = 1%2Flog%28w%2C+%28z%29%29 = 1%2F%28%281%2F60%29%29 = 60.


ANSWER.  log%28z%2C+%28w%29%29 = 60.

Solved.


T h a t ' s     a l l.


            +---------------------------------------------------------+
            |   It is how this problem  SHOULD  BE  SOLVED.   |
            +---------------------------------------------------------+


For your safety,  IGNORE  everything that  @MathLover1 wrote in her post.


This woman demonstrates extreme activity,  but solves many problems  INCORRECTLY,  or in a way,
which is not adequate and/or is not effective.

Unfortunately.

So,  be careful,  considering her contribution  (!)