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

HallsofIvy

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Jan 27, 2012
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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 \(\displaystyle log_2(2^{2x+2}+ 2^2)\).
 
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Eugenio

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Jul 4, 2021
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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.
 

Jomo

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Dec 30, 2014
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log base ab of ac=c/a.
This fact above will tell you exactly what the lhs equals.

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Since math is not a spectator sport we want you to be involved in solving your problem.
 
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