Logarithmic Function - Performance Problem

Quote:

Originally Posted by peekay

Been having trouble with this one, any help would be appreciated.

Log2x + Log4x = 5 (both the 2 and the 4 are subscripts)

So wouldn't you put it into exponential form and then solve for the trinomial?
So it would be like 2^2x +2^x-5=0 and then let a variable = 2^x?

From the change base formula you know:

$\log_4(x)=\dfrac{\ln(x)}{\ln(4)}=\dfrac{\ln(x)}{2 \ln(2)}=\dfrac12 \log_2(x)$

Therefore the given equation becomes:

$\log_2(x)+\dfrac12 \log_2(x)=5~\implies~\log_2(x)=\dfrac{10}3~\implie s~\boxed{x=2^{\frac{10}3} = \sqrt[3]{1024}}$