SOLUTION: Dear Sir/Madam, Please help me with this logarithm problem, If {{{b = log(3,x)}}}, find x in {{{log(b,(log(3,x^2)))=2}}}. note: the base in log3x is 3, the base in

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Dear Sir/Madam, Please help me with this logarithm problem, If {{{b = log(3,x)}}}, find x in {{{log(b,(log(3,x^2)))=2}}}. note: the base in log3x is 3, the base in      Log On


   

Question 895153: Dear Sir/Madam,

Please help me with this logarithm problem,
If b+=+log%283%2Cx%29, find x in log%28b%2C%28log%283%2Cx%5E2%29%29%29=2.


note: the base in log3x is 3, the base in logb is b and the base in log3x^2 is 3.

Thank you very much!
Found 2 solutions by Jc0110, MathTherapy:
Answer by Jc0110(165) About Me  (Show Source):
You can put this solution on YOUR website!
log%28b%2C%28log%283%2Cx%5E2%29%29%29=2
b+=+log%283%2Cx%29
log%283%2Cx%29=b
x=3%5Eb ... Substitute into log%28b%2C%28log%283%2Cx%5E2%29%29%29=2
log%28b%2C%28log%283%2C%283%5Eb%29%5E2%29%29%29=2
log%28b%2C%28log%283%2C3%5E%282b%29%29%29%29=2
log%28b%2C2b%28log%283%2C3%29%29%29=2
log%28b%2C2b%281%29%29=2
log%28b%2C2b%29=2
2b=b%5E2
b%5E2-2b=0
b%28b-2%29=0
b=0 ... b=0 is not acceptable.
or
%28b-2%29=0
b=2 ... b=2 is acceptable.
Substitute b=2 into x=3%5Eb
x=3%5E2
x=9

Answer by MathTherapy(10555) About Me  (Show Source):
You can put this solution on YOUR website!

Dear Sir/Madam,

Please help me with this logarithm problem,
If b+=+log%283%2Cx%29, find x in log%28b%2C%28log%283%2Cx%5E2%29%29%29=2


note: the base in log3x is 3, the base in logb is b and the base in log3x^2 is 3

Thank you very much!


log%28b%2C%28log%283%2C+x%5E2%29%29%29+=+2

log%283%2C+x%5E2%29 = 2log+%283%2C+x%29

Since log+%283%2C+x%29+=+b, then 2log+%283%2C+x%29, or log+%283%2C+x%5E2%29 = 2b

log%28b%2C+2b%29+=+2 ------ Substituting 2b for log+%283%2C+x%5E2%29 in log%28b%2C%28log%283%2C+x%5E2%29%29%29+=+2

b%5E2+=+2b

b%5E2+-+2b+=+0

b%28b+-+2%29+=+0

Thus, b = 2                 OR            b = 0 (ignore)



Now, since b+=+log%283%2C+x%29, and b = 2, then:

2+=+log+%283%2C+x%29

x+=+3%5E2, or highlight_green%28x+=+9%29