Why is

$\displaystyle 10^(lg{5})=5$

$\displaystyle e^(ln{5})=5$

so on and so forth?

Thanks.

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- Sep 7th 2010, 05:44 AM #1

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- Sep 7th 2010, 05:53 AM #2
u have a strange name

anyway

$\displaystyle log x = log_{10} x $ when u did not write the base it is 10

$\displaystyle log_a a = 1 $ in general

if u do not know what is the logarithm this is an example

suppose $\displaystyle x^a = b $ then $\displaystyle a = \log _x b $

so

to find $\displaystyle 10^{\log 5} $

let $\displaystyle x = 10^{\log_{10} 5} $

$\displaystyle \log _{10} x = \log _{10} 5 $ so $\displaystyle x = 5 $

second one same as the first one

$\displaystyle ln = \log _{e} $

- Sep 7th 2010, 06:09 AM #3

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- Sep 7th 2010, 06:13 AM #4

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- Sep 7th 2010, 07:56 AM #5

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Specifically, $\displaystyle log_a(x)$ or "logarithm to base a of x" is defined as the "inverse function" to $\displaystyle a^x$. That is, each "undoes" the other: $\displaystyle a^{log_a(x)}= x$ and $\displaystyle log_a(a^x)= x$.

Your "lg" is an abbreviation for the "common logarithm" or "logarithm base 10", $\displaystyle log_{10}(x)$ which is the inverse to $\displaystyle 10^x$. $\displaystyle lg(10^x)= log_{10}(10^x)= x$ for all x (and so for x= 5 $\displaystyle lg(10^5)= 5$). $\displaystyle 10^{lg(x)}= 10^{log_{10}(x)}= x$ for all positive x (and so for x= 5 $\displaystyle 5^{lg(5)}= 5$.

"ln" is an abbreviation for the "natural logarithm" or "logarithm base e" (e is about 2.718...) which is inverse to $\displaystyle e^x$. $\displaystyle ln(e^x)= log_e(e^x)= x$ for all x (and so for x= 5 $\displaystyle ln(e^5)= 5$. $\displaystyle e^{ln(x)}= x$ for all positive x (and so for x= 5 $\displaystyle e^{ln(x)}= x$.)

(Note the difference between "for all x" and "for positive x". The function "$\displaystyle a^x$ is only defined for positive a and always has a positive value- that is, $\displaystyle a^x$ is a function from the set of all real numbers to the set of all**positive**real numbers. It's inverse, $\displaystyle ln_a(x)$, is a function from the set of all positive real numbers to the set of real numbers.)

- Sep 7th 2010, 09:01 AM #6

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- Sep 7th 2010, 09:06 AM #7

- Sep 7th 2010, 01:01 PM #8

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- Sep 7th 2010, 01:27 PM #9