Logarithmic exponenent

• October 22nd 2011, 07:43 AM
Bashyboy
Logarithmic exponenent
How would I read an equation like this 6^(log_6 20)=20 ? How would I evaluate it? Why is it equal to 20?
• October 22nd 2011, 08:08 AM
Siron
Re: Logarithmic exponenent
In general $y=\ln(x) \Leftrightarrow e^{y}=x \Leftrightarrow e^{\ln(x)}=x$. Do you recognize this in your example?
You can also take the $\log_6$ of both sides and simplify with the logarithm rules.
• October 22nd 2011, 08:15 AM
Bashyboy
Re: Logarithmic exponenent
I am sorry, but I do not quite understand the defintion you provided. What do the arrows represent?
• October 22nd 2011, 08:20 AM
Siron
Re: Logarithmic exponenent
The arrows indicate that it's equivalent to write $y=\ln(x)$ or $e^{y}=x$ for example. But do you understand it now?
• October 22nd 2011, 08:27 AM
Bashyboy
Re: Logarithmic exponenent
I think I am beginning to. Why is it, though, that when the base of the logarithm in the exponent and the base of that logarithmic exponenent are equal, the answer is arguement? Is it because it is not an algebraic function, and is read differently because of that.
• October 22nd 2011, 09:06 AM
Siron
Re: Logarithmic exponenent
It's just a very handy proposition to work with logarithms. You'll see if you take the $\log_6$ of both sides then you get:
$\log_6(6^{\log_6(20)})=\log_6(20)$
$\Leftrightarrow \log_6(20)\cdot \log_6(6)=\log_6(20)$
$\Leftrightarrow \log_6(6)=1$
$\Leftrightarrow 6^1=6$

So indeed, they LHS and the RHS of the expression are equal.