Logs and Exponents

**Twenty80** · January 18th 2009, 11:22 AM

I'm a little stumped here, as we have a constant to the power of an exponent.. How would I simplify this:

$6^{\log_{2}n}$

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I would also appreciate it if someone could tell me if the answer I came up with for the following is correct:
Simplify the following:
$\log_{10}(10(\log_{10}n^3)100^{2n})$

My answer:
$\log_{10}(30\log_{10}n) + 4n$

I feel my answer isn't complete as I'm not too sure how to further simplify what's left of the addition operator.

**vincisonfire** · January 18th 2009, 03:22 PM

For the first one you have
$6^{log_2 n} = (2\cdot 3)^{log_2 n} = 2^{log_2 n} 3^{log_2 n} = n 3^{log_2 n}$
For the second
$\log_{10}(30\log_{10}n) + 4n = \log_{10}^2(n) + \log_{10}(3) + \log_{10}(10) + 4n = \log_{10}^2(n) + \log_{10}(3) + 1 + 4n$

**Mush** · January 18th 2009, 03:32 PM

Originally Posted by vincisonfire

For the first one you have
$6^{log_2 n} = (2\cdot 3)^{log_2 n} = 2^{log_2 n} 3^{log_2 n} = n 3^{log_2 n}$
For the second
$\log_{10}(30\log_{10}n) + 4n = \log_{10}^2(n) + \log_{10}(3) + \log_{10}(10) + 4n = \log_{10}^2(n) + \log_{10}(3) + 1 + 4n$

Alternatively:

$\log_{2}{n} = \log_{6}n \times \log_{2}6$

Hence:

$6^{\log_{6}n \times \log_{2}6}$

$(6^{\log_{6}n})^{\log_{2}6}$

$(n)^{\log_{2}6}$

$(n)^{\log_{2}(2\times3)}$

$(n)^{\log_{2}2 + \log_{2}3}$

$(n)^{1+ \log_{2}3}$

**vincisonfire** · January 18th 2009, 03:35 PM

Originally Posted by Mush

$(n)^{1+ \log_{2}3}$

Certainly a better answer.

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