SOLUTION: (7^log₈x)(x^log₉x) = 3969, find x.

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Question 1209562: (7^log₈x)(x^log₉x) = 3969,
find x.

Found 3 solutions by Edwin McCravy, ikleyn, mccravyedwin:
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
With a TI-84 and other technology it is easy to get an approximate answer.

An approximate answer I can read right off my trusty TI-84 calculator as

x = 15.545914

However, the teacher who assigned this problem is probably not teaching a course
on how to use technology to solve math problems.  So I think students who are
assigned problems like this are supposed to get EXACT answers, in terms of
standard functions, such as logarithms and exponential functions. So I'll strive
for an EXACT answer. It's not going to be easy.   

(7^log₈x)(x^log₉x) = 3969, 

%287%5Elog%283%2C%28x%29%29%29%28x%5Elog%289%2C%28x%29%29%29=3969

The left side is a strictly increasing function.

When x=9 the left side is

%287%5Elog%283%2C%289%29%29%29%2A%289%5Elog%289%2C%289%29%29%29

7%5E2%289%5E1%29=441

Thus x must be greater than 9, since the left side must increase all the way to
3969.

%287%5Elog%283%2C%28x%29%29%29%28x%5Elog%289%2C%28x%29%29%29=3969

%287%5Elog%283%2C%28x%29%29%29%28x%5Elog%289%2C%28x%29%29%29=7%5E2%2A3%5E4

Divide both sides by 72.

%287%5E%28log%283%2C%28x%29%29-2%29%29%28x%5Elog%289%2C%28x%29%29%29=3%5E4 

Take natural logs of both sides:

 

ln%287%5E%28log%283%2C%28x%29%29-2%29%29%2Bln%28x%5Elog%289%2C%28x%29%29%29%22%22=%22%224%2Aln%283%29

%28log%283%2C%28x%29%29-2%29ln%287%29%2B%28log%289%2C%28x%29%29%29ln%28x%29%22%22=%22%224%2Aln%283%29

Change to natural logs:

%28ln%28x%29%2Fln%283%29-2%29ln%287%29%2B%28ln%28x%29%2Fln%289%29%29ln%28x%29%22%22=%22%224%2Aln%283%29

Since ln%289%29=ln%283%5E2%29=2%2Aln%283%29,

%28ln%28x%29%2Fln%283%29-2%29ln%287%29%2B%28ln%28x%29%2F%282%2Aln%283%29%29%29ln%28x%29%22%22=%22%224%2Aln%283%29

let ln(x) = y,  ln(3) = a, and ln(7) = b

%28y%2Fa-2%29%2Ab%2B%28y%2F%282a%29%29y%22%22=%22%224a

%28by%29%2Fa-2b%2By%5E2%2F%282a%5E%22%22%29%22%22=%22%224a

2by-4ab%2By%5E2%22%22=%22%228a%5E2

y%5E2%2B2by-4ab-8a%5E2%22%22=%22%220

y%5E2%2B2by-4a%28b%2B2a%29=0

y+=+%28-2b+%2B-+sqrt%28+%282b%29%5E2-4%2A1%2A%28-4a%28b%2B2a%29%29%29+%29%2F%282%2A1%29+

y+=+%28-2b+%2B-+sqrt%28+4b%5E2%2B16a%28b%2B2a%29%29+%29%2F2+

y+=+%28-2b+%2B-+sqrt%28+4%28b%5E2%2B4a%28b%2B2a%29%29%29+%29%2F2+

y+=+%28-2b+%2B-+2sqrt%28+b%5E2%2B4ab%2B8a%5E2%29%29+%2F2+

y+=+-b+%2B-+sqrt%28b%5E2%2B4ab%2B8a%5E2%29++

ln%28x%29+=+-ln%287%29+%2B-+sqrt%28ln%287%29%5E2%2B4ln%283%29ln%287%29%2B8ln%283%29%5E2%29++

Since x > 9 then ln(x) > 2, so we must use the + sign.

x%22%22=%22%22

x%22%22=%22%22

Since e%5E%28-ln%287%29%29=1%5E%22%22%2Fe%5E%28ln%287%29%29=1%2F7

x%22%22=%22%22

That's the EXACT answer.

When you evaluate that with a calculator, you'll get x = 15.54591432.

Edwin

Answer by ikleyn(52779) About Me  (Show Source):
You can put this solution on YOUR website!
.
(7^log₈x)(x^log₉x) = 3969,
find x.
~~~~~~~~~~~~~~~~~~~~~~~


Edwin, in your post, you solved different equation comparing with the given in the post.

In the post, first parenthesis is 7 in degree log base 8 of x.

_____________________Not 7 in degree log base 3 of x, as it is written in your post.



Answer by mccravyedwin(407) About Me  (Show Source):
You can put this solution on YOUR website!
Thanks, Ikleyn.  You're right, now that I look more closely.  The 8 was so small,
I thought it was a 3.  I may redo it with 8.  But it will probably be harder to
get an exact answer since 8 and 9 are relatively prime. Using 8, the approximate
answer is x = 28.81741

Edwin