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Logarithms and Exponents

For problems 1 -3, use the algebraic definition of the logarithm to evaluate (simplify) the expression by hand and state the property number used. All work must be shown.

1. log8 (8^(x-1)
2. log4 ((1/16)^7)
3. log (0.0001)

For problems 4 - 6, solve for x exactly by hand.

4. log5 (x - 1) = 1
5. log2 (1/8) = x
6. log7 (1) = x - 2

For problems 7 and 8, simplify the following expressions completely using properties of logarithms. Keep all work on a line-by-line basis.

7. (3/2)log5(t6) - 2log5(t1/2) + 3log5(3) - (1/2)log5(81)
8. (20)log3(x1/4) - (2)log3(y) + (3)log3(y2/3) - (5/4)log3(x4)

For problems 9 and 10, solve the exponential equations. Obtain both an exact and approximate value. As a reminder, do not use your calculator until you have algebraically solved for x. Include and distinguish both exact and approximate values.

9. (5)e(3t - 1) = 85
10. (2)3x - 7 = 11

11. A company manufacturing surfboards has fixed costs of $300 per day and total costs of $5,100 per day at a daily output of 20 boards.

a) Assuming that the total cost per day, , is linearly related to the total output per day, x, write an expression for the cost function.

b) The average cost per board for an output of x boards is given by . Find the average cost function.

c) On your own, graph the average cost function on your calculator for Do NOT include the graph here.

d) What does the average cost per board tend to as production increases? Briefly justify your answer using complete sentence(s).

Extra Credit
Read pages 126 - 128. Then try problem 103 on page 121 on your own (nothing to show here) and confirm your solution. Now try to answer question 104:

Refer to Table 3, Page 121, question 104:

a) Find a logarithmic regression model y=a+b ln(x) for the Total Production.
b) Calculate to the nearest million, the production in 2010. (Show the steps for evaluating the model for a given value x.)

Solution Preview

For problems 1 -3, use the algebraic definition of the logarithm to evaluate (simplify) the expression by hand and state the property number used. All work must be shown.

1.

loga(b) = c means the same as ac = b. The first step to all of these problems will be writing the logarithms in that form.

So, log8(8x-1) = ?, is the same as 8? = 8x-1. That means to solve this problem, we want to figure out what the ? must be so that 8? = 8x-1. From looking at that last equation, you can see that ? = x - 1.

Therefore, log8(8x-1) = x - 1.

The answer is x - 1.

Note: This problem is an example of how logarithms and exponents "cancel out". When you have a number raised to an exponent (23 for example), and you take the logarithm with the same base as the original number (log2(23)), you are left with what was in the exponent (log2(23) = 3). See another example of this in the working of problem 9:

e(3t - 1) = 17
ln(e(3t - 1)) = ln(17)
3t - 1 = ln17 (the ln and the e on the left side canceled out)

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2.

log4((1/16)7) = ?, is the same as 4? = (1/16)7.

Try to make the right side of that equation look more like the left side:

4? = (1/16)7
4? = ((1/4)2)7
4? = (1/4)14
4? = ((4)-1)14
4? = (4)-14

? = -14

Therefore, log4((1/16)7) = -14.

Another way to do this is to use the properties of logarithms (see question 7 for the "rules").

log4((1/16)7)
7log4(1/16) using rule (1)
7[log4(1) - log4(16)] using rule (3)
7[0 - log4(16)] because log(1) = 0 no matter what the base is (see question 6)
-7log4(42) factor out the -1 and write 16 ...

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