SOLUTION: Solve for x: log base 4 (x^2 + 3x) - log base 4 (x +5) = 1 I know that I am supposed to do this; 4^1 = x^2 + 3x ----------- -

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Solve for x: log base 4 (x^2 + 3x) - log base 4 (x +5) = 1 I know that I am supposed to do this; 4^1 = x^2 + 3x ----------- -       Log On


   

Question 448342: Solve for x: log base 4 (x^2 + 3x) - log base 4 (x +5) = 1
I know that I am supposed to do this; 4^1 = x^2 + 3x
-------------
x +5
HOwever I am not entirely sure that this is the correct method
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First use the idea that log%28b%2C%28A%29%29-log%28b%2C%28B%29%29=log%28b%2C%28A%2FB%29%29


So the left side should condense into log%284%2C%28%28x%5E2%2B3x%29%2F%28x%2B5%29%29%29 where in this case A=x%5E2%2B3x, B=x%2B5 and b=4



So the equation simplifies to log%284%2C%28%28x%5E2%2B3x%29%2F%28x%2B5%29%29%29=1


Now you can use the property you were referring to and you'll get

4%5E1=%28x%5E2%2B3x%29%2F%28x%2B5%29


4=%28x%5E2%2B3x%29%2F%28x%2B5%29 Simplify


4%28x%2B5%29=x%5E2%2B3x Multiply both sides by x%2B5


4x%2B20=x%5E2%2B3x Distribute.


0=x%5E2%2B3x-4x-20 Get everything to the right side.


0=x%5E2-x-20 Combine like terms.


Notice that the quadratic x%5E2-x-20 is in the form of Ax%5E2%2BBx%2BC where A=1, B=-1, and C=-20


Let's use the quadratic formula to solve for "x":


x+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 Start with the quadratic formula


x+=+%28-%28-1%29+%2B-+sqrt%28+%28-1%29%5E2-4%281%29%28-20%29+%29%29%2F%282%281%29%29 Plug in A=1, B=-1, and C=-20


x+=+%281+%2B-+sqrt%28+%28-1%29%5E2-4%281%29%28-20%29+%29%29%2F%282%281%29%29 Negate -1 to get 1.


x+=+%281+%2B-+sqrt%28+1-4%281%29%28-20%29+%29%29%2F%282%281%29%29 Square -1 to get 1.


x+=+%281+%2B-+sqrt%28+1--80+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-20%29 to get -80


x+=+%281+%2B-+sqrt%28+1%2B80+%29%29%2F%282%281%29%29 Rewrite sqrt%281--80%29 as sqrt%281%2B80%29


x+=+%281+%2B-+sqrt%28+81+%29%29%2F%282%281%29%29 Add 1 to 80 to get 81


x+=+%281+%2B-+sqrt%28+81+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%281+%2B-+9%29%2F%282%29 Take the square root of 81 to get 9.


x+=+%281+%2B+9%29%2F%282%29 or x+=+%281+-+9%29%2F%282%29 Break up the expression.


x+=+%2810%29%2F%282%29 or x+=++%28-8%29%2F%282%29 Combine like terms.


x+=+5 or x+=+-4 Simplify.


So the possible solutions are x+=+5 or x+=+-4


However, we need to check them to verify that they are actually solutions. To do so, just plug in the values of x and simplify to get an identity.


It turns out that both solutions check out. So the two solutions are x+=+5 or x+=+-4