Answer to Question #259253 in Algebra for bri

Question

a)Solve the following:

i) 2^x/2-16=1/32       ii) log₈(x-2)+log₈(x)=1

b) The demand curve for a swivel chair is given by p=4000(3^-q) dollars per swivel chair, where p is the price and q is the quantity, in thousands of swivel chair, demanded at that price. What quantity will be demanded if the price per swivel chair is $256.60?

 

Answer 1

(a)

(i)

$2 ^{\frac {x} {2} -16} = \frac {1} {32} \\$

$2 ^{\frac {x} {2} -16} = \frac {1} {2^5} \\$

$2 ^{\frac {x} {2} -16} = 2^ {-5} \\ Equating \ the \ indices; \\ \frac{x} {2} - 16 = -5 \\ Grouping \ like \ terms \ (i.e \ adding \ 16 \ to \ both \ sides) \\ \frac{x} {2} - 16 + 16 = -5 + 16 \\ \frac{x} {2} = 11 \\ Multiplying \ both \ sides \ by \ 2 \\ \frac{x} {2} \times 2 = 11 \times 2 \\ x = 22$

(ii)

$\log_8 (x -2) + \log_8 (x) = 1 \\ According \ to \ the \ rule \ of \ logarithm, \\ \log A + \log B = \log (A \times B) \\ \therefore \log_8 (x -2) + \log_8 (x) = 1 \\ \log_8 x(x -2) = 1 \\ \log_8 x(x -2) = \log_8 8 \\ Equating\ the \ logarithm, \\ x(x -2) = 8 \\ x^2 - 2x = 8 \\ x^2 - 2x - 8 = 0 \\ Solving \ using \ the \ quadratic \ formula; \\ x = \frac {-b \pm \sqrt {b^2 - 4ac}} {2a} \\ a = 1, b = -2, c = -8 \\ x = \frac {-(-2) \pm \sqrt {(-2)^2 - 4(1)(-8)}} {2 \times 1} \\ x = \frac {2 \pm \sqrt {4 + 32}} {2} \\ x = \frac {2 \pm \sqrt {36}} {2} \\ x = \frac {2 \pm 6} {2} \\ x = \frac {2 + 6} {2} \ or \ x = \frac {2 - 6} {2} \\ x = \frac {8} {2} \ or \ x = \frac {-4} {2} \\ x = 4 \ or \ x = -2$

since logarithm of negative number is not possible, (i.e., $x \isin \mathbb{N}$)

$\therefore x = 4$

(b)

$p = 4000(3^{-q}) \\ \frac {p} {4000} = 3^{-q}$

Taking the logarithm to base 3 of both sides;

$\log_3 \frac {p} {4000} = \log_3 3^{-q} \\ \log_3 \frac {p} {4000} = -q \log_3 3 \\ since \ \log_3 3 =1 \\ \log_3 \frac {p} {4000} = -q \\ So, \\ q = - \log_3 \frac {p} {4000} \\ \therefore$

The quantity that will be demanded if the price per swivel chair is $256.60 is calculated by:

$q = - \log_3 \frac {256.60} {4000}$

changing logarithm of base three to base ten;

$\log_a b = \frac {\log_c (b)} {\log_c (a)}$

$q = - \log_3 \frac {256.60} {4000} = - \frac {\log_{10} \frac {256.60} {4000}} {\log_{10} 3}$

$q = - \frac {(-1.1928)} {0.4771}$

$q = 2.5001 \\ q \approx 2.5$

Therefore, 2.5 thousand swivel chairs will be demanded with the price of $256.60