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Math 115: Sample Test 2 - Logarithms and Exponentials - Prof. J. S. Carter, Exams of Mathematics

University of South Alabama (USA)Mathematics
Prof. J. S. Carter

Math 115: sample test 2 from spring 2004 focusing on logarithms and exponentials. It includes problems on computing logarithms, simplifying expressions using logarithmic rules, identifying asymptotic behavior of functions, sketching graphs, and more.

Typology: Exams

Pre 2010

Uploaded on 08/18/2009

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koofers-user-rzk 🇺🇸

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Math 115 Carter Sample Test 2 Spring 2004
1. Compute the following (Hint: using a calculator will slow you down):
(a) log6(216)
(b) log2(1
2096 )
(c) log1/3(729)
(d) log343 (7)
2. Use the rules of logarithms and exponentials to simplify the expressions.
(a) log12 (x)−log12 (2x−1) + 3 log12 (x−4)
(b) 153x+2151−3x
(c) 367log367 (453x−4)
(d) ln (e42)
3. Given that logA(2) = 0.3869,logA(3) = 0.6131, logA(7) = 1.0860, and logA(10) =
1.2851, compute
(a) logA(12)
(b) logA(2/3)
(c) logA(72)
(d) logA(49)
(e) logA(28)
4. There will be at least four graphs that are similar to these. For each of the following graph
f(x) indicating horizontal, vertical and other asymptotic behavior. Write solutions to
each of the the inequalities f(x)≤0, f(x)<0, 0 <f(x), and 0 ≥f(x)
(a) f(x)= x2
x2−4
(b) f(x)= x
x2−4
(c) f(x)= 1
x2−4
(d) f(x)= x3
x2−4
(e) f(x)= 1
(x−4)(x−2)
1
pf2

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Math 115 Carter Sample Test 2 Spring 2004

  1. Compute the following (Hint: using a calculator will slow you down):

(a) log 6 (216) (b) log 2 ( 20961 ) (c) log 1 / 3 (729) (d) log 343 (7)

  1. Use the rules of logarithms and exponentials to simplify the expressions.

(a) log 12 (x) − log 12 (2x − 1) + 3 log 12 (x − 4) (b) 15^3 x+2 151 −^3 x (c) 367log^367 (453x−4) (d) ln (e^42 )

  1. Given that logA (2) = 0. 3869 , logA (3) = 0.6131, logA (7) = 1.0860, and logA (10) = 1 .2851, compute

(a) logA (12) (b) logA (2/3) (c) logA (72) (d) logA (49) (e) logA (28)

  1. There will be at least four graphs that are similar to these. For each of the following graph f (x) indicating horizontal, vertical and other asymptotic behavior. Write solutions to each of the the inequalities f (x) ≤ 0, f (x) < 0, 0 < f (x), and 0 ≥ f (x)

(a) f (x) = (^) xx (^2) −^24 (b) f (x) = (^) x 2 x− 4 (c) f (x) = (^) x (^21) − 4 (d) f (x) = (^) xx (^2) −^34 (e) f (x) = (^) (x−4)(^1 x−2)

1

(f) f (x) = (^) x (^2) +3x^3 x− 2 (g) f (x) = (x − 2)(x + 3)(x − 4) (h) f (x) = (x − 4)(x + 2)(x + 1)^2

  1. Sketch the graphs of

(a) y = 17^4 x−^5 (b) y = ln (x − 2) (c) y = 4 ln (x − 3) + 3

  1. Compute the difference quotient f^ (x+h h)− f^ (x) for the function f (x) = ex.
  2. In the analogy between addition and multiplication, the analogous statement to “mul- tiplication distributes over addition” is “exponentiation distributes over multiplica- tion.” Given that the former law of arithmetic can be expressed in the equation, (a + b)c = ac + bc, express the latter law as an equation.
  3. Find all solutions to the equation:

(a) log 10 (x) + log 10 (x + 3) = 1 (b) log 5 (x + 4) + log 5 (x − 4) = 2

  1. The half-life of peanuts in the Tappa Tappa Keg Frat house is 8 days. The frat bros buy 10 barrels of peanuts on Sept. 1. How long will it take until 86% of the peanuts are GONE? How many barrels of peanuts will remain on Sept 30?
  2. Compute the amount of time it will take for $600,000 to grow to 2.5 million dollars assuming continuously compounded interest at 7.5 % per year.