1. Draw a quick sketch of the graph of log(x)
Pick x values and find the corresponding y values. Here are a few points: (1,0) that is.. when x = 1, then log(1) = 0 since 10^0 = 1, (10,1), that is log(10) = 1 since 10^1 = 10.. (100, 2) What about when x<1? How about x = .1? 10^-1 = 1/10 or .1 so log(.1) = -1. You cannot have a value for log(0) since there is no power you can raise 10 to in order to get 0, but you can get mighty close. For example, log(.00001) would give you a y value of -5, since 10^-5 would be 1/100000 or .00001 You can use gcalc.net to see this function
3. 213(base 4 ) = _________ base 10
since this is base 4, the 3 is in the ones position, the 1 is in the 4^1 or 4 position, and the 2 is in the 4^2 or 16 position. 213, then, in base 4 is equal to 3 + 1(4) + 2(16) or 39 in base 10
5. 104 (base 10 ) = __________ base 2
in base 2, the place values are powers of 2. They double each place over, and start at 1. So… the place values are 1,2,4,8,16,32,64,128,….. but instead of going left to right, we go right to left in our numbering system, so lets flip it: 128, 64, 32, 16, 8, 4, 2, 1 We want to express 104 in base 2. What is the biggest place value that is less than 104? 64.. that is, we could not have a 1 in the 128 position, since 128 is already too much. If we have a 1 in the 64 position, we have 64 accounted for of the 104 we need to express. Put a 1 in the 64 position (the 7th position over from the right) and subtract 64 from 104 to get 40. Repeat the process.. the first number smaller than 40 is the 32.. put a 1 in the 32 position and subtract 32 from 40 to get 8. Notice you do not have enough to put a 1 in the 16 position, but you do have 8 to put a 1 in the 8 position. now you are done: Your answer, long format is 1(64) 1(32) 0(16) 1(8) 0(4) 0(2) 0(1) or just 1101000
7. log (base 7 ) 88 = __________ in exponential format
Your base is 7, so the question is: what power do you have to raise 7 to in order to get 88? The answer can be written this way: 7^x = 88
9. 3^2.5 = 15.588 is _________ in log format
Here the base is 3. You are raising 3 to the 2.5 power to get 15.588. In log format, this can be written as: log(base 3 ) 15.588 = 2.5
11. Without using a calculator, estimate log (base 6 ) 89 and tell your process.
What power do you have to raise 6 to in order to get 89? 6^1 = 6, which is too small. 6^2 is 36 and still too small. 6^3 is 216, which is too big, so the answer is between 2 and 3. It will be 2.(something..) Since 89 is closer to 36 than it is to 216, you would be tempted to say something like 2. 3 or even 2.2. Remember, though, that this is in a logarithm scale. Half way in terms of the power is really about 70 percent of the way. so 2.5 would be much closer to 216 (70 percent closer) than it would be to 36. A good guess, then, would be 2.4. 2.7 would be too high. If you check using a calculator, take 6^2.4 power and see you are close to 89… 6^2.5 is almost perfect.
13. Rewrite using the log product rule: log(21)
log(xy) = log(x) + log(y) So check.. what are the factors of 21. That is.. what 2 numbers multiplied together get 21… this would be 3(7) So, log(21) is the same as log(3(7)) which is log(3) + log(7)
15. Rewrite using the log quotient rule: log(10/3)
When dividing, you subtract your logs.. log(x/y) = log(x) – log(y) In this case, you have log(10) – log(3). But.. log(10) can be written as log(2(5)) or log(2) + log(5). So the final answer should be: (log(2) + log(5) ) – log(3)
17. Use both log quotient and product rule: log (3x/2y)
Similar to #13 and #15 above, the answer is: (log(3) + log(x)) – (log(2) + log(y))
19. Rewrite using a single log function: (log(x) + log(3) ) – log(y)
Here you are combining instead of breaking apart.. log(x) + log(3) would be the same as log(3x). Log(3x) – log(y) would be the same as log(3x/y)
21. Rewrite using the log power rule: log(x^5) = _______
If you take the log of something to a power, you have the number of the power times the base, which means you have, in this case, 5 log(x)s, which you can rewrite as 5 log(x)
23. Solve using the log power rule: 21^x = 100
take the log of both sides to get: log(21^x) = log(100) rewrite as: x log(21) = log(100) Divide both sides by log(21) to get: x = log(100)/log(21) You can find the decimal number using your calculator to get about 1.51
25: Have an answer to the following from your group:
a. Why do we need logarithms?
Logs allow you to solve for a variable that is in the exponent position
b. Explain how logs and exponents are related
The log of a number (base 10) would be what exponent you would have to raise 10 to in order to get that number
c. How would you find log(base8 ) 59 using a calculator (no log 8 key)
Find the whole number part of the log. In this case, 8^1 is 8 and 8^2 is 64, so the log(base 8 ) of 59 is between 1 and 2. Using the exponent feature of your calculator, make a guess as to what decimal between 1 an 2 might be close. In this case, 59 is pretty close to 64, so try maybe 1.8 as the exponent. Find 8^1.8. If this answer is over 59, make the the exponent smaller, it 1.7. Continue until you have it bracketed.
d. Why is log(xy) equal to log(x) + log(y)
This works because when you multiply numbers with the same base and different exponents, you keep the base the same and add the exponents. For more detail, let a = log(x) and b = log(y) then 10^a = x and 10^b=y You want log(xy) Using substitution, this is the same as log(10^a(10^b)) But this is the same as log(10^(a+b)) which means you need to raise 10 to the a + b power to get xy. This gives you log(x) + log(y)
e. Why is log(x/y) equal to log(x) – log(y)
Similare to d above. You will have 10^a/10^b which is the same as 10^(a-b)
f. Why is log x^t equal to t(logx)
Take an example: log(x^5) this is the same, from d above, as log(x) + log(x) + log(x) + log(x) + log(x), since x^5 is just x(x)(x)(x)(x) You have 5 log(x). In the question above, you have t multiples of x, which becomes t log(x’s) added together, which gives you tlog(x)