how do i solve this equation log(4, x^2 + 3 * x) - log(4, x + 5) = 1?

Question

log4 ( x^2 + 3x) - log4 (x + 5) = 1

Answer 1

Assuming those are log-base-4...

log4(x^2 + 3x) - log4 (x + 5) = 1
log4(x^2 + 3x) = log4(x+5) + 1

However, 1 = log4(4), so...

log4 (x^2 + 3x) = log4(x+5) + log4(4)

And since log(a) + log(b) = log(ab)...

log4 (x^2 + 3x) = log4(4 * (x+5))

Now we can remove the logs, since if log(a) = log(b) then a=b:

x^2 + 3x = 4x + 20

Move everything over to one side:

x^2 - x - 20 = 0

20 = 4*5 which have a difference of 1, so this is:

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

x = 5, x = -4 are the solutions.

Going back to check:

When x=5:
log4(x^2 + 3x) - log4(x + 5) = 1
log4(5^2 + 3*5) - log4(5+5) =? 1
log4(25 + 15) - log4(10) =? 1
log4(40) - log4(10) =? 1
log4(40 / 10) =? 1
log4(4) =? 1
1 = 1

When x=-4:
log4(x^2 + 3x) - log4(x + 5) = 1
log4((-4)^2 + 3*(-4)) - log4((-4)+5) =? 1
log4(16 - 12) - log4(1) =? 1
log4(4) - 0 =? 1
1 = 1

Both answers check out.

Answer 2

log4 ( x^2 + 3x) - log4 (x + 5) = 1
log4 ( x^2 + 3x) - log4 (x + 5) = log 4(0)
x^2 + 3x - (x + 5) = 0
x^2 + 3x - x - 5 = 0
x^2 + 2x - 5 = 0
then u can solve it by find the roots of the quad equation

Answer 3

question a million) while including logs with elementary bases, multiply their arguments into one yet another: log(2x^2 + x) = a million Convert to exponential variety, log has a organic base of 10 so while you're no longer given a base anticipate that its 10 (base of 10, exponent of a million): 2x^2 + x = 10^a million 2x^2 + x = 10 flow each and every thing to a minimum of one ingredient: 2x^2 + x - 10 = 0 element it out: (x - 2) (2x + 5) = 0 x = 2, -5/2 notice: in case you flow decrease back to the start of the equation you will see that -5/2 includes taking the log a destructive quantity, you may no longer take the log of a destructive quantity or 0. hence -5/2 isn't a valid answer. x = 2 --------------------- question 2) while subtracting logs with elementary bases, divide their arguments from one yet another: log4(x-2)/(x-3) = 2 Convert to exponential variety (base of four, exponent of two): (x - 2) / (x - 3) = 4^2 (x - 2) / (x - 3) = sixteen Multiply the two aspects via (x - 3) to cancel off the denominator: x - 2 = 16x - 40 8 upload 40 8 to the two aspects: x + 40 six = 16x Subtract x from the two aspects: 40 six = 15x Divide the two aspects via 15: x = 40 six/15