Can someone please help me with a logarithmic problem!! solve for x: log4 (x^2+3X) - log4 (x+5) =1?

Answer 1

Using the fact that log a - log b = log a/b:

log4 (x² + 3x)/(x + 5) = 1

Then getting rid of the logs by exponentation by 4:

(x² + 3x)/(x + 5) = 4^1 = 4

Multiply both sides by (x + 5):

x² + 3x = 4(x + 5) = 4x + 20

Combine on one side:

x² -x - 20 = 0

Factor:

(x - 5)(x + 4) = 0

x - 5 = 0, so x = 5, or
x + 4 = 0, so x = -4

Check:

log4 (5² + 3(5)) - log4 (5 + 5) = 1
log4 (25 + 15) - log4 10 = 1
log4 40 - log4 10 = 1
log4 4(10) - log4 10 = 1 (then using log 4(10) = log 4 + log 10)
log4 4 + log4 10 - log4 10 = 1
log4 4 = 1 = 1, check!

log4 (4² +3(-4)) - log4(-4 + 5) = 1
log4 4 - log4 1 = 1
1 - 0 = 1, check!

Answer 2

log4 (x^2+3X) - log4 (x+5) =1
<=> log4 (x^2+3x) = 1 + log4 (x+5)
<=> log4 (x^2+3x) = log4 [4(x+5)]
<=> x^2 + 3x = 4x + 20 and x > -5
<=> x^2 - x -20 = 0 and x > -5
<=> x = 5 or x = -4 and x > -5
so x =5 or x = -4

Answer 3

Rule: log?(x) - log?(y) = log?(x/y) log?(x) = y ? x = a^y log?(x²+3x) - log?(x+5) = a million log?((x²+3x) / (x+5)) = a million (x²+3x) / (x+5) = 4¹ = 4 x² + 3x = 4(x+5) = 4x + 20 x² - x + 20 = 0 (x - 5)(x + 4) = 0 x = -4, 5 Neither of those solutions violate the area of the unique function, the argument of a log can not be damaging or 0. as a result they are the two ultimate.

Answer 4

Remember that log(a) - log(b) = log(a/b); the rule holds regardless of the base.

log[4]( (x² + 3x) / (x+5) ) = 1

Raise both sides upon the base 4; the log disappears on the left.
(x² + 3x) / (x+5) = 4^1 = 4

Multiply both sides by (x+5) to get it out of the denominator:
x² +3x = 4(x +5)

Expand out:
x² + 3x = 4x + 20

Put everything on the left:
x² - x - 20 = 0

Factor:
(x - 5)(x + 4) = 0

The solutions are x = 5 and x = -4.

Double-checking:
log[4] (5² + 3(5)) - log[4](5 + 5) =?
log[4] (40) - log[4] (10) =?
log[4] (40/10) =?
log[4](4) = 1 <-- check

log[4] ((-4)² + 3(-4)) - log[4](-4 + 5) =?
log[4] (16 - 12) - log[4](1) =?
log[4] (4) - log[4](1) =?
log[4](4) - 0 =?
log[4](4) = 1 <-- check

So the answers are:
x = 5 or x = -4

Answer 5

log4(x²+3x) - log4(x+5) = 1
log4[(x²+3x)/(x+5)] =1

Apply inverse logarithmic function:
[(x²+3x)/(x+5)] =4^1
(x²+3x)/(x+5)=4
x²+3x = 4(x+5)
x²-x-20=0

Solving the last quadratic equation, we get:

x1 = 5
x2 = -4

That's it!

Good luck!

Answer 6

(x^2+3x)/(x+5)=4

x^2+3x=4x+20
x^2-x-20=0
(x-5)(x+4)=0
x=5 or -4