What is the value of x? Find it from the picture given below:

[Abu Sayeed Mahi]

What is the value of x?

 

[lex]

Is the topic related to numerical solutions of equations?
Otherwise, I wonder is there a mistake in the statement.

 

[HallsofIvy]

A logarithm is the inverse function to the exponential. For any a and x, $\displaystyle log_a(a^x)= x$.

$\displaystyle 8= 2^3$ so $\displaystyle \sqrt{8}= 2^{3/2}$ and $\displaystyle \frac{1}{\sqrt{8}}= \left(\frac{1}{\sqrt{2}}\right)^3$.

$\displaystyle 4= 2^2$ so $\displaystyle 4^{x+1}= (2^2)^{x+1}= 2^{2(x+1)}= 2^{2x+2}$ so $\displaystyle log_2(4^{x+ 1})=2x+ 2$.

It is the last logarithm that is the problem. There is no way to simplify $\displaystyle log_2(2^{2x+2}+ 2^2)$.

 

[Eugenio]

The LHS is 3. The RHS is 6 when x = 0 and 0 when x = -1. The solution is between -1 and 0. You need a numerical approach.

 

[Steven G]

log base a^b of a^c=c/a.
This fact above will tell you exactly what the lhs equals.

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