Simplifying more Logarithms? =\?
(a) log_10 (10^(1/2))
(b) log_10 ( 1/(10^x))
(c) 2 log_10 sqrt(x) + 3 log_10 x^(1/3)
Best Answer goes to the person that explains it step by step.. Damn this one is hard =(
- cj kLv 41 decade agoFavorite Answer
a) log_10 (10^(1/2)) asks 10 to what power is 10 to the 1/2 power. Well clearly the answer is 1/2.
b) Ten to what power is (1/(10^x)). The fraction can be rewritten as 10^(-x), so the answer is clearly (-x)
c) For this you have to know that A log (b) is equal to log (b to the A power). So the first half becomes log_10 of the sqrt x to the second power, the sqrt of a number that is then raised to the second power is the number itself. This also works for x^1/3 raised to the 3rd power is also x. So you now have log_10 x + log_10 x, which can be combined into log_10 x^2.
- 1 decade ago
Your first key concept is that log_10(10^b) = b for any b. That's the basic idea of a logarithm.
(a) 1/2, directly from the above
To get (b), remember the definition of a negative exponent: x^-b = 1/(x^b).
Therefore, log_10(1/(10^x)) = log_10(10^-x) = -x. (The last step goes back to the first concept above.)
To get (c), you need to know that log_10(x^n) = nlog_10(x).
2log_10sqrt(x) + 3log_10(x^1/3) = 2log_10(x^1/2) + 3log_10(x^1/3)
= 2(1/2)log_10(x) + 3(1/3)log_10(x) = log_10(x) + log_10(x) = 2log_10(x)
- kumorifoxLv 71 decade ago
Remember these rules about logs.
The log of a power equals the power times the log.
The log of a product is equal to the sum of the logs.
The log of a quotient is equal to the difference of the logs.
The log of a number to its own base is equal to one.
The log of 1 to any base is equal to 0.
The log of 0 and of negative numbers is undefined.
(A) log(10) 10^1/2
1/2 log(10) 10 (use log of a power)
1/2×1 = 1/2 (use log to its own base = 1)
(B) log(10) 1/10^x
log(10) 1 - log(10) 10^x (use log of quotient rule)
0 - x log(10) 10 (use log of 1 = 0 and log of power)
-x (use log of number to own base)
(C) 2 log(10) √x + 3 log(10) x^1/3
Remember that √x = x^1/2
2 log(10) x^1/2 + 3 log(10) x^1/3
log(10) x^[(1/2)(2)] + log(10) x^[(1/3)(3)] (reverse log of power rule: a number times a log equals the log of a power)
log(10) x + log(10) x = 2 log(10) x = log(10) x²
Hope that helps a little.