SOLUTION: {{{ log( 2, a ) + log( a, 4)= 3}}} Determine the value of 'a' Thanks

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: {{{ log( 2, a ) + log( a, 4)= 3}}} Determine the value of 'a' Thanks      Log On


   

Question 323586: +log%28+2%2C+a+%29+%2B+log%28+a%2C+4%29=+3
Determine the value of 'a'
Thanks
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Change of base formula: log%28b%2C%28x%29%29=ln%28x%29%2Fln%28b%29


log%282%2C%28a%29%29%2Blog%28a%2C%284%29%29=3 Start with the given equation.


ln%28a%29%2Fln%282%29%2Bln%284%29%2Fln%28a%29=3 Use the change of base formula (shown above)


ln%28a%29%2Aln%28a%29%2Bln%284%29%2Aln%282%29=3%2Aln%282%29%2Aln%28a%29 Multiply EVERY term by the LCD ln%282%29%2Aln%28a%29 to clear out the fractions.



Now let's make z=ln%28a%29. This then means that we get


z%2Az%2Bln%284%29%2Aln%282%29=3%2Aln%282%29%2Az


z%5E2%2Bln%284%29%2Aln%282%29=3%2Aln%282%29%2Az Rewrite z%2Az to get z%5E2


z%5E2-3%2Aln%282%29%2Az%2Bln%284%29%2Aln%282%29=0 Subtract 3%2Aln%282%29%2Az from both sides.


It may be hard to see, but the equation is now in the form Az%5E2%2BBz%2BC=0. To solve for 'z', use the quadratic formula


z+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29+


In this case, A=1, B=-3%2Aln%282%29 and C=ln%284%29%2Aln%282%29. Plug all this into the formula to get






Square -3%2Aln%282%29 to get 9%2A%28ln%282%29%29%5E2 and multiply


z+=+%283%2Aln%282%29+%2B-+sqrt%28ln%282%29%289%2Aln%282%29-4%2Aln%284%29%29+%29%29%2F2+ Factor out the GCF ln%282%29 from the terms under the radical.



z+=+%283%2Aln%282%29+%2B-+sqrt%28ln%282%29%28ln%282%5E9%29-ln%284%5E4%29%29+%29%29%2F2+ Use the identity y%2Aln%28x%29=ln%28x%5Ey%29


z+=+%283%2Aln%282%29+%2B-+sqrt%28ln%282%29%28ln%28512%29-ln%28256%29%29+%29%29%2F2+ Raise 2 to the 9th power to get 512 and raise 4 to the 4th power to get 256.


z+=+%283%2Aln%282%29+%2B-+sqrt%28ln%282%29%28ln%28512%2F256%29%29+%29%29%2F2+ Combine the logs using the identity ln%28A%29-ln%28B%29=ln%28A%2FB%29


z+=+%283%2Aln%282%29+%2B-+sqrt%28ln%282%29%2Aln%282%29+%29%29%2F2+ Divide


z+=+%283%2Aln%282%29+%2B-+sqrt%28%28ln%282%29%29%5E2+%29%29%2F2+ Rewrite ln%282%29%2Aln%282%29 as %28ln%282%29%29%5E2


z+=+%283%2Aln%282%29+%2B-+ln%282%29%29%2F2+ Take the square root of %28ln%282%29%29%5E2 to get ln%282%29


The plus/minus then breaks down into two equations: z+=+%283%2Aln%282%29+%2B+ln%282%29%29%2F2+ or z+=+%283%2Aln%282%29+-+ln%282%29%29%2F2+


z+=+%283%2Aln%282%29+%2B+ln%282%29%29%2F2+ Start with the first equation.


z+=+%284%2Aln%282%29%29%2F2+ Add


z+=+2%2Aln%282%29+ Reduce.


z+=+ln%282%5E2%29+ Use the identity y%2Aln%28x%29=ln%28x%5Ey%29


z+=+ln%284%29+ Square 2 to get 4.


Since we let z=ln%28a%29, this means that ln%284%29=ln%28a%29 and that a=4 is one solution.


Now move onto the second equation


z+=+%283%2Aln%282%29+-+ln%282%29%29%2F2+ Start with the second equation.


z+=+%282%2Aln%282%29+%29%2F2+ Subtract


z+=+ln%282%29+ Reduce.


Again, we let z=ln%28a%29, meaning that ln%28a%29=ln%282%29 which implies that a=2


So a=2 is the other solution. We know that there are only two solutions here because we're dealing with a quadratic.


====================================================
Answer:

So the two solutions are a=2 or a=4


I'll leave the check for you to do.