6^x=4 ? i have a sudden black out and i can't remember the rule for when solving equations like this one.. to complicate things i'm not allowed to answer with x=log4/log6

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- Feb 11th 2008, 12:10 PM #1

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- Feb 11th 2008, 01:03 PM #2

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- Feb 11th 2008, 01:05 PM #3

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- Feb 11th 2008, 01:08 PM #4

- Feb 11th 2008, 01:10 PM #5

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- Feb 11th 2008, 01:11 PM #6

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- Feb 11th 2008, 01:13 PM #7

- Feb 11th 2008, 01:14 PM #8

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- Feb 11th 2008, 01:19 PM #9
__Method1:__

Initial equation

$\displaystyle 6^x=4$

Take the log of both sides (it doesn't matter what base you use)

$\displaystyle log(6^x)=log(4)$

One of the properties of logs is that if they are the log of a number to a power, they are the same as the power times the log of the number. Meaning $\displaystyle log(6^x)=xlog(6)$ so

$\displaystyle xlog(6)=log(4)$

Divide both sides by log(6)

$\displaystyle x=\frac {log(4)}{log(6)}$

__Method2__

Initial equation

$\displaystyle 6^x=4$

Take log base 6 of both sides

$\displaystyle log_6(6^x) = log_6(4)$

Simlify

$\displaystyle x = log_6(4)$

Use change of base formula to rewrite in a format you can plug into a calculator. Change of base formula says $\displaystyle log_a(b) = \frac{log_c(b)}{log_c(a)}$ This works for any value of c. Typically the value chosen is 10 or e, as log base ten and log base e (natural log) are the two most common logs, and are programmed into any decent calculator.

$\displaystyle x = \frac{log(4)}{log(6)}$

- Feb 11th 2008, 01:21 PM #10

- Feb 11th 2008, 01:28 PM #11

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- Feb 11th 2008, 01:31 PM #12

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- Feb 11th 2008, 01:32 PM #13
Then the answer he is looking for is

$\displaystyle x = \frac{log(4)}{log(6)} \approx$ 0.77370561446908317374049227693595

Which is not exact, and I would seem to require a calculator to find.

It seems most likely to me that they are looking for $\displaystyle x=\frac{log(4)}{log(6)}$ or $\displaystyle x=log_6(4)$ as these values are exact, and one of the other requirements was that he could not use a calculator.

- Feb 11th 2008, 01:34 PM #14

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- Feb 11th 2008, 01:35 PM #15

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