SOLUTION: Please help me Solve for all values of x, to the nearest 10th. log4(2x−7)+log4(3x−3)=4

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Please help me Solve for all values of x, to the nearest 10th. log4(2x−7)+log4(3x−3)=4       Log On


   

Question 617078: Please help me
Solve for all values of x, to the nearest 10th.
log4(2x−7)+log4(3x−3)=4
Found 3 solutions by ewatrrr, jsmallt9, josmiceli:
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
 

Hi



 4^4 = (2x-7)(3x-3)

  6x^2 -27x + 21 - 256 = 0

  3x^2 - 27x -235 = 0

x+=+%2827+%2B-+sqrt%28+6873%29%29%2F%2812%29+

 x: -4.7,  9.2  (rounded to the nearest tenth)

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Assuming these are base 4 logarithms...
log%284%2C+%282x-7%29%29%2Blog%284%2C+%283x-3%29%29=4
We want the equation in the form
log(expression) = other-expression
So we will use a property of logarithms, log%28a%2C+%28p%29%29%2Blog%28a%2C+%28q%29%29+=+log%284%2C+%28p%2Aq%29%29, to combine the logarithms:
log%284%2C+%28%282x-7%29%283x-3%29%29%29=4

Now that the equation is in the desired form, the next step is to rewrite the equation in exponential form. In general log%28a%2C+%28p%29%29+=+q is equivalent to a%5Eq+=+p. Using this pattern on our equation we get:
4%5E4+=+%282x-7%29%283x-3%29
which simplifies to:
256+=+6x%5E2-27x%2B21

Now that the variable is out of the logarithm, we can solve for it. This is a quadratic equation so we want one side to be zero. Subtracting 256 from each side:
0+=+6x%5E2-27x-235
Next we factor or use the Quadratic Formula. This will not factor so we must use the formula:
x+=+%28-%28-27%29+%2B-+sqrt%28%28-27%29%5E2+-+4%286%29%28-235%29%29%29%2F2%286%29
Simplifying...
x+=+%28-%28-27%29+%2B-+sqrt%28729+-+4%286%29%28-235%29%29%29%2F2%286%29
x+=+%28-%28-27%29+%2B-+sqrt%28729+-+%28-5640%29%29%29%2F2%286%29
x+=+%28-%28-27%29+%2B-+sqrt%28729+-+%28-5640%29%29%29%2F2%286%29
x+=+%28-%28-27%29+%2B-+sqrt%286369%29%29%2F2%286%29
x+=+%2827+%2B-+sqrt%286369%29%29%2F12
So x+=+%2827+%2B+sqrt%286369%29%29%2F12 or x+=+%2827+-+sqrt%286369%29%29%2F12

Since you were asked for decimal answers, get out your calculator and round off your answer to the nearest tenth.

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Use the general rule:
+log%28a%29+%2B+log%28b%29+=+log%28a%2Ab%29+
Also, express the right side as a log to base 4
+log%284%2C2x-7%29+%2B+log%284%2C3x-3%29+=+4+
+log%284%2C+%282x-7%29%2A%283x-3%29%29+=+log%284%2C256%29+
+%282x-7%29%2A%283x-3%29+=+256+
+3%2A%282x-7%29%2A%28x-1%29+=+256+
+3%2A%28+2x%5E2+-+7x+-+2x+%2B+7+%29+=+256+
+2x%5E2+-+9x+%2B+7+=+256%2F3+
+2x%5E2+-+9x+-+235%2F3+
+x+=+%28-b+%2B-+sqrt%28+b%5E2+-+4%2Aa%2Ac+%29%29+%2F+%282%2Aa%29+
+a+=+2+
+b+=+-9+
+c+=+-235%2F3+
+x+=+%28-%28-9%29+%2B-+sqrt%28+%28-9%29%5E2+-+4%2A2%2A%28-235%2F3%29+%29%29+%2F+%282%2A2%29+
+x+=+%28+9+%2B-+sqrt%28+81+%2B+1880%2F3%29+%29%29+%2F+4+
+x+=+%28+9+%2B-+sqrt%28+243%2F3+%2B+1880%2F3%29+%29%29+%2F+4+
+x+=+%28+9+%2B-+sqrt%28+2123%2F3%29+%29%29+%2F+4+
+x+=+%28+9+%2B-+sqrt%28+707.6667%29+%29%29+%2F+4+
+x+=+%28+9+%2B+26.602+%29%2F4+
+x+=+35.602%2F4+
+x+=+8.9005+
To nearest tenth, +x+=+8.9+
check answer:
+log%284%2C2x-7%29+%2B+log%284%2C3x-3%29+=+4+
+log%284%2C2%2A8.9-7%29+%2B+log%284%2C3%2A8.9-3%29+=+4+
+log%284%2C17.8-7%29+%2B+log%284%2C26.7-3%29+=+4+
+log%284%2C10.8%29+%2B+log%284%2C23.7%29+=+4+
+log%284%2C+10.8%2A23.7%29+=+log%284%2C256%29+
+log%284%2C+255.96%29+=+log%284%2C256%29+
Looks close enough
-----------------
Note that x had to be positive because a log
with a positive base (4), can't result in a negative number