Precalculus question

Question

Find the largest value of  that satisfies:

x = __?

Answer 1

The difference of the logarithms is the logarithm of a quotient .

So,

log ₂ x² - log ₂ (x+5) = 6 --------- (1) can be re-written as

log ₂ ( x² / (x+5) ) = 6 ------------(2)

By the definition of the logarithm, this is

x ² = 2⁶

(x+5)

x ² = 64 Multiplying both sides by (x+5),

(x+5)

x² = 64 (x+5) . Writing this as a standard quadratic equation,

x² - 64x - 320 -0

using the quadratic formula to solve, we obtain

x = 64 ± 73.32 ( you are highly encouraged to double check and let me know )

2

Considering the positive value, x = 68.66.

BUT WE CANNOT STOP HERE!! Substitute the x value in the original problem in (2) to check if this is the answer.

Is log ₂ ( 68.66² / (68.66+5) ) = 6?

log ₂ (64.00052) = 6 ?

6 = 6 ? Yes.

Therefore x = 68.66.

Note: I want you to find out why the negative value of x is rejected .

Answer 2

about 58.5

log2(x^2) - log2(x+5) = 6

logx(x^2/(x+5) = 6

2^6 = 64 = x^2/(x+5)

64(x+5) = x^2

x^2 -64x - 320 = 0

x=32 + or - (1/2)sqr(64^2 +4(320) = 32 + 26.5 = about 58.5 (ignore the negative root if you want max x)

Answer 3

log₂ [x² / (x + 5)] = 6

2⁶ = x² / (x + 5)

Can you solve for s and answer?