SOLUTION: If: log₂x + log₄x + log₁₆x = 21/4 find x.

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Question 1206690: If:
log₂x + log₄x + log₁₆x = 21/4
find x.
Found 5 solutions by MathLover1, greenestamps, math_tutor2020, MathTherapy, Edwin McCravy:
Answer by MathLover1(20855) About Me  (Show Source):
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log%282%2Cx%29+%2B+log%284%2Cx+%29%2B+log%2816%2Cx%29+=+21%2F4....change to base 10

log%28x%29%2F+log%282%29%2B+log%28x+%29%2Flog%284%29%2B+log%28x%29%2F+log%2816%29=+21%2F4...use common denominator










14%28log%282%29%29%5E2%2Alog%28x%29=42%28log%282%29%29%5E3...both sides divide by 14%28log%282%29%29%5E2+

log%28x%29=3log%282%29

log%28x%29=log%282%5E3%29...if log same, then

x=2%5E3

x=8


Answer by greenestamps(13327) About Me  (Show Source):
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Ouch!!!

The solution method shown by the other tutor is valid, but WAY more work than needed....

log%282%2C%28x%29%29%2Blog%284%2C%28x%29%29%2Blog%2816%2C%28x%29%29=21%2F4

The bases of the logarithms are all powers of 2, making for a quick and easy path to the answer.

16=2%5E4, so log%282%2C%28x%29%29=4%2Alog%2816%2C%28x%29%29
16=4%5E2, so log%284%2C%28x%29%29=2%2Alog%2816%2C%28x%29%29

So



Then

7%2Alog%2816%2C%28x%29%29=21%2F4
log%2816%2C%28x%29%29=3%2F4
x=16%5E%283%2F4%29=8

ANSWER: 8

Answer by math_tutor2020(3835) About Me  (Show Source):
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Answer: x = 8

Work Shown

w = log(x)
v = log(2)

2v = 2*log(2) = log(2^2) = log(4)
4v = 4*log(2) = log(2^4) = log(16)
log%282%2C%28x%29%29+%2B+log%284%2C%28x%29%29+%2B+log%2816%2C%28x%29%29+=+21%2F4

Change of base rule

w%2Fv%2Bw%2F%282v%29%2Bw%2F%284v%29+=+21%2F4

%28w%2Fv%29%281%2B1%2F2%2B1%2F4%29+=+21%2F4

%28w%2Fv%29%287%2F4%29+=+21%2F4

w%2Fv+=+%2821%2F4%29%2A%284%2F7%29

w%2Fv+=+3

w+=+3v

log%28%28x%29%29+=+3%2Alog%28%282%29%29

log%28%28x%29%29+=+log%28%282%5E3%29%29

log%28%28x%29%29+=+log%28%288%29%29

x+=+8

Answer by MathTherapy(10806) About Me  (Show Source):
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If:
log₂x + log₄x + log₁₆x = 21/4
find x. 

 
  ---- Changing 
   
 ---- Multiplying by LCD, 4
                  matrix%281%2C3%2C+7log%282%2C+%28x%29%29%2C+%22=%22%2C+21%29 
                   matrix%281%2C3%2C+log%282%2C+%28x%29%29%2C+%22=%22%2C+3%29 ----- Dividing both sides by 7
                        ----- Converting to EXPONENTIAL form

Answer by Edwin McCravy(20077) About Me  (Show Source):
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log%282%2Cx%29+%2B+log%284%2Cx%29+%2B+log%2816%2Cx%29+=+21%2F4
find x. 

matrix%281%2C2%2CLet%2C+x+=+16%5Ey%29

log%282%2C16%5Ey%29+%2B+log%284%2C16%5Ey%29+%2B+log%2816%2C16%5Ey%29+=+21%2F4

y%2Alog%282%2C16%29+%2B+y%2Alog%284%2C16%29+%2B+y%2Alog%2816%2C16%29+=+21%2F4

y%2Alog%282%2C2%5E4%29+%2B+y%2Alog%284%2C4%5E2%29+%2B+y%2Alog%2816%2C16%29+=+21%2F4

Since log%28a%2Ca%5En%29+=+n,

4y%2B2y%2B1y=21%2F4

7y=21%2F4

y=3%2F4

x=16%5Ey=16%5E%28%223%2F4%22%29=%282%5E4%29%5E%28%223%2F4%22%29=+2%5E3=8

Edwin