# 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)

_=subscript

Best Answer goes to the person that explains it step by step.. Damn this one is hard =(

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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.

• Simplify Log10 10

• 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)

• 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.