Question

1. solve the logarithmic equation: $log_{2}(x) + log_{2}(x - 2) = 3$.

a. $x = 4$
b. $x = 3$
c. $x = 4, x = -2$
d. $x = 5$

1. if an investment grows continuously according to the formula $a = pe^{rt}$, and you want to double your money with a rate $r = 0.05$, which equation solves for the time $t$

a. $t = 0,05\ln(2)$
b. $2 = 1.05^{t}$
c. $2 = e^{0.05t}$
d. $1 = e^{0.05t}$

1. convert $150^{\circ}$ to radians.

a. $\frac{2\pi}{3}$
b. $\frac{5\pi}{6}$

Explanation:

Question 9

Step1: Combine log terms

$\log_2(x(x-2)) = 3$

Step2: Convert to exponential form

$x(x-2) = 2^3 = 8$

Step3: Expand and rearrange to quadratic

$x^2 - 2x - 8 = 0$

Step4: Factor quadratic

$(x-4)(x+2) = 0$

Step5: Check domain (x>2)

Reject $x=-2$ since $\log_2(-2)$ is undefined.

Step1: Set $A=2P$ (double the money)

$2P = Pe^{rt}$

Step2: Divide both sides by $P$

$2 = e^{rt}$

Step3: Substitute $r=0.05$

$2 = e^{0.05t}$

Step1: Use degree to radian formula

$\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$

Step2: Substitute 150 degrees

$150 \times \frac{\pi}{180} = \frac{5\pi}{6}$

Answer:

A. $x = 4$

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Question 10