Assume that , , and . Use these values and the properties of logarithms to find the value. Do not use a calculator.
A) 3.1661
B) 2.2588
C) 1.3929
D) 2.1583
E) 2.5814

Question

Assume that  ,  , and  . Use these values and the properties of logarithms to find the value. Do not use a calculator. 
A) 3.1661 
B) 2.2588 
C) 1.3929 
D) 2.1583 
E) 2.5814

Quizplus · Accepted Answer

Each term is a hexadecimal representation of a unique ID, except for the final ID we are solving for. The last digit of each ID is the same, implying that they are all related to each other by a logarithm with base 16 (since we are working in hexadecimal).

Let the first ID be written as x, the second as x + a, and the third as x + 2a, where a is some unknown value. Then we have:

log16(x + a) - log16(x) = log16(11ec1e8e_2c1e_86eb_be9b_734beb359589_TBX8967_11) - log16(11ec1e8e_2c1e_86ea_be9b_0d0ce2ea7707_TBX8967_11) = log16(3.1661)

log16(x + 2a) - log16(x + a) = log16(11ec1e8e_2c1e_86ec_be9b_037fc58e17d7_TBX8967_11) - log16(11ec1e8e_2c1e_86eb_be9b_734beb359589_TBX8967_11) = log16(2.1583)

Therefore, we have:

log16(x + 2a) - log16(x) = log16(x + 2a) - log16(x + a) + log16(x + a) - log16(x) = log16(2.1583) + log16(3.1661) = log16(8.6349)

Simplifying, we get:

log16((x + 2a)/x) = log16(8.6349)

Therefore:

(x + 2a)/x = 8.6349

Solving for a, we get:

a = x*(8.6349-1)/2.6349

Using the second given ID, we can solve for x:

log16(x + a) - log16(x) = log16(11ec1e8e_2c1e_86eb_be9b_734beb359589_TBX8967_11) - log16(11ec1e8e_2c1e_86ea_be9b_0d0ce2ea7707_TBX8967_11) = log16(3.1661)

Simplifying, we get:

log16(1 + a/x) = log16(3.1661)

Therefore:

1 + a/x = 3.1661

Solving for x, we get:

x = a/(3.1661-1)

Now we can solve for the final ID using the third given ID:

log16(x + 2a) = log16(11ec1e8e_2c1e_86ec_be9b_037fc58e17d7_TBX8967_11)

Substituting in our values and simplifying, we get:

log16(5.2689a) = log16(2.5814)

Therefore:

a = 2.5814/5.2689

Using this value of a, we can solve for x and find the final ID:

x = a/(3.1661-1) = 0.6588

Final ID: 11ec1e8e_2c1e_86ed_be9b_9f3852b64819_TBX8967_11.

Assume That , , and