SOLUTION: Solve: 2 * log--base4 (x) + 3 * log--base8 (x)=10

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Question 481624: Solve:
2 * log--base4 (x) + 3 * log--base8 (x)=10
Found 3 solutions by lwsshak3, MathTherapy, greenestamps:
Answer by lwsshak3(11628) About Me  (Show Source):
You can put this solution on YOUR website!
Solve:
2 * log--base4 (x) + 3 * log--base8 (x)=10
**
2log4(x)+3log8(x)=10
Change to base 10
2log(x)/log4+3log(x)/log8=10
log(x)((2/log4)+(3/log8))=10
log(x)(3.3219+3.3219)=10
log(x)(6.6439)=10
log(x)=10/6.6439=1.5051
x=31.9963=32

Answer by MathTherapy(10837) About Me  (Show Source):
You can put this solution on YOUR website!
Solve:

2 * log--base4 (x) + 3 * log--base8 (x)=10
******************************************
Respondent @wsshak3(11628) came up with a decimal value for x, but this problem clearly has an INTEGER-solution, as
seen below!! While both are close to each other, the approximated decimal-answer doesn't QUITE MAKE the equation true,
but ALMOST TRUE, while the INTEGER-value for x, below, does.

                   2log%284%2C+%28x%29%29+%2B+3log+%288%2C+%28x%29%29+=+10
    
 ---- Converting ALL logs to base 2
       2%28log+%282%2C+%28x%29%29%29%2F2+%2B+3%28log+%282%2C+%28x%29%29%29%2F3+=+10
   
                      log%282%2C+%28x%29%29+%2B+log+%282%2C+%28x%29%29+=+10
                                    2log%282%2C+%28x%29%29+=+10 
                                      log+%282%2C+%28x%29%29+=+10%2F2
                                      log+%282%2C+%28x%29%29+=+5
                                                 x+=+2%5E5 ----- Converting to EXPONENTIAL form
                                               highlight%28x%29+=+highlight%2832%29%29

Answer by greenestamps(13351) About Me  (Show Source):
You can put this solution on YOUR website!


2log%284%2C%28x%29%29%2B3log%288%2C%28x%29%29=10

Use nlog%28b%2C%28a%29%29=log%28b%2C%28a%5En%29%29 on both log expressions.

log%284%2C%28x%5E2%29%29%2Blog%288%2C%28x%5E3%29%29=10

Use log%28b%5En%2C%28a%5En%29%29=log%28b%2C%28a%29%29 on both log expressions.

log%282%2C%28x%29%29%2Blog%282%2C%28x%29%29=10

2log%282%2C%28x%29%29=10

log%282%2C%28x%29%29=5

x=2%5E5=32

ANSWER: x = 32