SOLUTION: Hi!! What is log base 4 of X - log base 16 of (x+3) = 1/2

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Question 1082504: Hi!! What is log base 4 of X - log base 16 of (x+3) = 1/2
Found 3 solutions by josgarithmetic, MathLover1, MathTherapy:
Answer by josgarithmetic(39718) About Me  (Show Source):
You can put this solution on YOUR website!
log base 4 of X - log base 16 of (x+3) = 1/2

log%284%2Cx%29-log%2816%2C%28x%2B3%29%29=1%2F2

log%284%2Cx%29-%28log%284%2C%28x%2B3%29%29%2Flog%284%2C16%29%29=1%2F2, using Change of Base formula

log%284%2Cx%29-%28log%284%2C%28x%2B3%29%29%2Flog%284%2C%284%5E2%29%29%29=1%2F2

log%284%2Cx%29-%28log%284%2C%28x%2B3%29%29%2F%282%2Alog%284%2C4%29%29%29=1%2F2

log%284%2Cx%29-%28log%284%2C%28x%2B3%29%29%2F2%29=1%2F2

2log%284%2Cx%29-log%284%2C%28x%2B3%29%29=1

log%284%2Cx%5E2%29-log%284%2C%28x%2B3%29%29=1

log%284%2C%28%28x%5E2%29%2F%28x%2B3%29%29%29=1

Next, put into exponential form,... and solve for x.

Answer by MathLover1(20854) About Me  (Show Source):
You can put this solution on YOUR website!


log%284%2Cx%29+-+log%2816%2C%28x%2B3%29%29+=+1%2F2

log%28x%29%2Flog%284%29+-+log%28%28x%2B3%29%29%2Flog%2816%29+=+1%2F2

log%28x%29%2Flog%282%5E2%29+-+log%28%28x%2B3%29%29%2Flog%282%5E4%29+=+1%2F2

log%28x%29%2F2log%282%29+-+log%28%28x%2B3%29%29%2F4log%282%29+=+1%2F2 .........both side multiply by 2

2log%28x%29%2F2log%282%29+-+2log%28%28x%2B3%29%29%2F4log%282%29+=+2%281%2F2%29

log%28x%29%2Flog%282%29+-+log%28%28x%2B3%29%29%2F2log%282%29+=+1 ....common denominator 2log%282%29

%282log%28x%29+-+log%28%28x%2B3%29%29%29%2F2log%282%29+=+1

2log%28x%29+-+log%28%28x%2B3%29%29+=+2log%282%29

log%28x%5E2%29+-+log%28%28x%2B3%29%29+=+log%282%5E2%29

log%28x%5E2%2F%28x%2B3%29%29+=+log%284%29

x%5E2%2F%28x%2B3%29+=+4

x%5E2+=+4%28x%2B3%29

x%5E2+=+4x%2B12

x%5E2+-4x-12=0

x%5E2+%2B2x-6x-12=0

%28x%5E2-6x%29+%2B%282x-12%29=0

x%28x-6%29+%2B2%28x-6%29=0

%28x+%2B2%29%28x-6%29=0

solutions:

%28x+%2B2%29=0=>x=-2.....disregard negative solution

%28x-6%29=0=>x=6-> your solution







Answer by MathTherapy(10699) About Me  (Show Source):
You can put this solution on YOUR website!
Hi!! What is log base 4 of X - log base 16 of (x+3) = 1/2

 log+%284%2C+%28x%29%29+-+log+%2816%2C+%28x+%2B+3%29%29+=+1%2F2
2%2Alog+%282%2C+%28x%29%29+-+log+%282%2C+%28x+%2B+3%29%29+=+2 ----- Converting from base 4 and base 16 to base 2 
log+%282%2C+%28x%5E2%29%29+-+log+%282%2C+%28x+%2B+3%29%29+=+2 ----- Applying a%2Alog+%28b%2C+%28c%29%29+=+log+%28b%2C+%28c%5Ea%29%29
       +log+%28highlight%282%29%2C+%28x%5E2%2F%28x+%2B+3%29%29%29+=+highlight_green%282%29 ---- Applying log+%28a%2C+%28b%29%29+-+log+%28a%2C+%28c%29%29+=+log+%28a%2C+%28b%2Fc%29%29 
              x%5E2%2F%28x+%2B+3%29+=+highlight%282%29%5Ehighlight_green%282%29 --- Converting to EXPONENTIAL form
              x%5E2%2F%28x+%2B+3%29+=+4
                 x%5E2+=+4%28x+%2B+3%29 ----- Cross-multiplying
                 x%5E2+=+4x+%2B+12
         x%5E2+-+4x+-+12+=+0
     (x - 6)(x + 2) = 0 ----- Factoring trinomial
x - 6 = 0      OR      (x + 2) = 0 ---- Equating each factor to 0
   x = 6      OR       x = - 2 (IGNORE, as this is an EXTRANEOUS solution!

So, only ONE (1) solution applies here, so accept NO OTHER.