Logs and Exponentials 1.
You have deposited $1000 in a money-­‐market account that earns 8 percent annual interest. Assuming no withdrawals or additional deposits are made, calculate how much money will be in the account one year later; two years later; three years later; t years later. 2.
Rewrite each equation so that it has the form “x = . . .” Please do not use “solve.” 5
3
1/5
3
(a) x = a (b) x = a (c) (1 + x)
15.6
−2
= 2.0 (d) x = a 3.
In order that a $10000 investment grow to $20000 in seven years, what must be the annual rate of interest? Seven years could be called the doubling time for this investment. Notice that it is being assumed that the interest is compounded. 4.
The population of Grand Fenwick has been increasing at the rate of 2.4 percent per year. It has just reached 5280 (a milestone). What will the population be after ten years? after t years? After how many years will the population be 10560? 5.
On one window on your graphing calc or on desmos, graph the equations y = 3 , y = 2 , y = 1.024 , and y =(1/2) . What do x
graphs of the form y = b have in common? How do they differ? 6.
On one system of coordinate axes, graph the equations y = 2 , y = (3)2 , y = (0.4)2 , and y = (−3)2 . What do all graphs x
x
y = k ⋅ 2 x have in common? How do they differ? 7.
x
Make up a context for the equation y = 5000(1.005) . x
x
x
x
x
x
8.
(Continuation) Find the value of x that makes y = 12500. Find the value of x that makes y = 2000. Interpret these answers in the context you chose. 9.
Write each of the following numbers as a power of 10. You should not need a calculator. 10 (e) 100 10 (a) 1000 (b) 1000000 (c) 0.01 (d)
10. Using your calculator, write 1997 as a power of 10. In contrast to the preceding question, this one is asking for an irrational exponent! Using your answer (but no calculator), also write 1
1997
as a power of 10. 11. (Continuation) Turn your calculator on, type LOG, (or press the LOG key), type 1997, and press ENTER. Compare the displayed value with the first of your two previous answers. 12. For each of the following, type LOG followed by the given number, and press ENTER. Interpret the results. By the way, “log” is short for logarithm, to be discussed soon. (a) 1000 (b) 1000000 (c) 0.01 (d)
10 (e) 100 10 13. Using the LOG function of your calculator, solve each of the following for x: (a) 10
x
= 3 (b) 10 x = 300 (c) 10 x = 9 (d) 10 x = 3−1 (e) 10 x = 3 You should see a few patterns in your answers — try to describe them. 14. Rewrite a. the logarithmic equation 4 = log 10000 as an exponential equation; b.
3.3004...
the exponential equation 10
()
= 1997 as a logarithmic equation. (
)t
15. The function p defined by p t = 3960 1.02 describes the population of Dilcue, North Dakota t years after it was founded. a. Find the founding population. b. At what annual rate has the population of Dilcue been growing? c.
Calculate p ( 65)
. p ( 64 )
16. (Continuation) Solve the equation p(t ) = 77218. What is the meaning of your answer? 17. Without using your calculator, solve each of the following equations: x
x
x
(a) 8 = 32 (b) 27 = 243 (c) 1000 = 100000 18. Explain why all three equations have the same solution. = p is called the base-­‐10 logarithm of p, expressed as x = log10 p , or 4
simply x = log p . For example, 10 = 10000 means that 4 is the base-­‐10 logarithm of 10000, or 4 = log10000 4 = log 19. Given a positive number p , the solution to 10
x
10000. The LOG function on your calculator provides immediate access to such numerical information. Using your calculator for confirmation, and remembering that logarithms are exponents, explain why it is predictable that a. log64 is three times log 4 ; b.
log12 is the sum of log 3 and log 4 ; c.
log0.02 and log50 differ only in sign. 20. You now know how to calculate logarithms by using 10 as a common base. Use this method to evaluate the following. Notice those for which a calculator is not necessary. (a) log85 (b) log58 (c) log5√5 (d) log1.0052.5 (e) log3(1/9) t
21. You have come to associate a function such as p(t) = 450(1.08) with the size of something that is growing (exponentially) at a fixed rate. Could such an interpretation be made for the function d
(t ) = 450 ⋅ 2t ? Explain. 22. Evaluate the following log expressions (A) log1 (B) log 2 32 (C) log 9 3 (D) log 1
2
2 (F) log 2540 (G) log 0.111 23. Rewrite a. the logarithmic equation: 4 = log 10000 as an exponential equation; b.
the exponential equation: 10
1.2787536
= 19 as a logarithmic equation. (E) log10 60 24. Rewrite the log equations in exponential form, and vice versa. (a) log10 10 = 1 (b) log 2 16 = 4 ⎛ 1⎞
1
(e) ⎜ ⎟ =
⎝ 2 ⎠ 16
(c) log a b = c (f) wm = k 4
(d) 5 = 125 3
x
25. What if the base of an exponential equation isn’t 10 ? One way of solving an equation like 1.02 = 3 is to use your calculator’s LOG function to rewrite the equation in the power of 10 form. a. Press LOG, enter 1.02. What is the result? Remember, this means that 1.02 is equal to 10, raised to that power. b.
Press LOG, enter 3. What is the result? Remember, this means that 3 is equal to 10, raised to that power. x
So this means that we can re-­‐write 1.02 = 3 as (10
nice property for us!!) 0.0086 x
) = 10
0.4771
, and we now have the property, 10 log x = x . (This will be a very 26. (Continuation) You have now calculated the logarithm of 3 using the base 1.02, for which log1.023 is the usual notation. The usual ways of reading log1.023 are “log base 1.02 of 3” or “log 3, base 1.02”, or “the base-­‐1.02 logarithm of 3”, or “log to the base 1.02 of 3.” Because many calculators do not have a button devoted to base-­‐1.02 logarithms, we have to convert the equation into a quotient of two base-­‐10 logarithms. For example: 1.02 = 3
x
log1.02 = log 3
x
x log1.02 = log 3 (What do you know about exponents that lets us do this step?)
x=
log 3
log1.02
x
27. Solve 2 = 1000. In other words, find log21000, the base-­‐2 logarithm of 1000. 28. You now know how to calculate logarithms by using 10 as a common base. Use this method to evaluate the following. Notice those for which a calculator is not necessary. (a) log85 (b) log58 (c) log5√5 (d) log1.0052.5 (e) log3(1/9) 29. The formula A ( t ) =
P (1+ kr ) is used for investments that compound a certain number of times per year. What does kt
that mean and what does each term stand for? 30. What would this represent? A (1) = 100
(1+ 1.00k )k(1) Compute this for k = {4;12; 365;525,600} k
1+ 1k ) . (
k→∞
31. Use your calculator or desmos to evaluate lim