PLEASE HELP ME WITH THESE QUESTION AS MUCH AS POSSIBLE!!
Evaluate without a calculator. Give exact answers:
a) log(log(10)) = ?
b) square root (log(100)) - log(square root(100)) = ?
c) log((square root 10^3)(square root 10^5)(square root 10) = ?
d) 1000^log(3) = ?
e) log(1/log(10square root 10)) = ?
a) log(log10)
= log 1 = 0
b) √log100 - log(√100)
= √2 - log10
=√2 - 1
c) log(√10^3√10^5√10)
= log √10^9
= log 10^(9/2)
= (9/2)log10 = (9/1)(1) = 9/2
d) 1000^log3
= ((10)^3)^log3
= 10 ^(3log3)
= 10^(log27)
= 27
e) log (1/log(10√10)
= log 1 - log(10√10)
= 0 - log(10√10)
= -log(10√10)
= -(log10 + log(√10)
= -(1 + 1/2_
= -3/2
a) To evaluate log(log(10)), we first need to evaluate log(10). The logarithm of 10 to the base 10 is 1. Now, taking the logarithm base 10 of 1, we have log(1) = 0.
So, log(log(10)) = 0.
b) To evaluate square root(log(100)) - log(square root(100)), we begin by evaluating log(100). The logarithm of 100 to the base 10 is 2.
Now, square root(log(100)) becomes square root(2), and square root(100) is 10.
Plugging in these values, we have: square root(2) - log(10).
But log(10) = 1, so substituting this: square root(2) - 1.
Therefore, square root(log(100)) - log(square root(100)) is equal to square root(2) - 1.
c) To evaluate log((square root(10^3))(square root(10^5))(square root(10))), we can simplify the expression inside the logarithm first.
(square root(10^3))(square root(10^5))(square root(10)) = (10^(3/2))(10^(5/2))(√10) = 10^((3+5)/2)√10 = 10^4 √10.
Now, taking the logarithm base 10 of 10^4 √10, we have log(10^4 √10) = log(10^4) + log(√10) = 4 + log(10) = 4 + 1 = 5.
Therefore, log((square root(10^3))(square root(10^5))(square root(10))) = 5.
d) To evaluate 1000^log(3), we can rewrite 1000 as 10^3.
So, the expression becomes (10^3)^log(3) = 10^(3*log(3)).
To evaluate 3*log(3), we first calculate log(3). Assuming base 10, log(3) is approximately 0.477.
Multiplying this value by 3, we get 1.431.
Therefore, 1000^log(3) is equal to 10^1.431, which is approximately 24,244.13.
e) To evaluate log(1/log(10 * square root(10))), we can simplify the expression inside the logarithm first.
10 * square root(10) = 10^(1/2 + 1) = 10^(3/2).
So, the expression becomes log(1/log(10^(3/2))).
Now, taking the inverse logarithm of 10^(3/2), we have: log(10^(3/2)) = 3/2.
Next, taking the inverse logarithm of 1/(3/2), we have log(1/(3/2)) = log(2/3).
Therefore, log(1/log(10 * square root(10))) = log(2/3).