+118609 Fin the product of all real solutions to \(x^{2 - \log_4(x)} = \pi\)
This turned out to be a really cool question!!
\(x^{2 - \log_4(x)} = \pi\\ log_4[x^{2 - \log_4(x)} ]=log_4[ \pi]\\ (2 - \log_4x) log_4x=log_4 \pi\\ 2log_4x - (\log_4x)^2=log_4 \pi\\ (\log_4x)^2-2(log_4x)+log_4 \pi=0\\ Let A=log_4x\\ A^2-2A+log_4\pi=0\\ A=\frac{2\pm\sqrt{4-4log_4\pi}}{2}\\ A=1\pm\sqrt{1-log_4\pi}\\ sub\; back\;again\\ log_4x=1\pm\sqrt{1-log_4\pi}\\ 4^{log_4x}=4^{1\pm\sqrt{1-log_4\pi}}\\ x=4^{1\pm\sqrt{1-log_4\pi}}\\~\\ \text{So the product of the only two values of x that make this true is}\\ 4^{1+\sqrt{1-log_4\pi}}\times 4^{1-\sqrt{1-log_4\pi}}\\ =4^{1+\sqrt{1-log_4\pi}+1-\sqrt{1-log_4\pi}}\\ =4^2\\ =16 \)
Mathematica 11 Home Edition gives the following 2 real values for x with no explanation!!
x =2.2425 and x =7.1348
2.2425 x 7.1348 =~ 16
+118609 I could not work out anything any more useful either.
Here are the 2 points on the graph
+118609 Fin the product of all real solutions to \(x^{2 - \log_4(x)} = \pi\)
This turned out to be a really cool question!!
\(x^{2 - \log_4(x)} = \pi\\ log_4[x^{2 - \log_4(x)} ]=log_4[ \pi]\\ (2 - \log_4x) log_4x=log_4 \pi\\ 2log_4x - (\log_4x)^2=log_4 \pi\\ (\log_4x)^2-2(log_4x)+log_4 \pi=0\\ Let A=log_4x\\ A^2-2A+log_4\pi=0\\ A=\frac{2\pm\sqrt{4-4log_4\pi}}{2}\\ A=1\pm\sqrt{1-log_4\pi}\\ sub\; back\;again\\ log_4x=1\pm\sqrt{1-log_4\pi}\\ 4^{log_4x}=4^{1\pm\sqrt{1-log_4\pi}}\\ x=4^{1\pm\sqrt{1-log_4\pi}}\\~\\ \text{So the product of the only two values of x that make this true is}\\ 4^{1+\sqrt{1-log_4\pi}}\times 4^{1-\sqrt{1-log_4\pi}}\\ =4^{1+\sqrt{1-log_4\pi}+1-\sqrt{1-log_4\pi}}\\ =4^2\\ =16 \)
Solve for x:
log(x) (2 - log(x)/log(4)) = log(π)
Expand out terms of the left hand side:
2 log(x) - (log^2(x))/log(4) = log(π)
Multiply both sides by -log(4):
log^2(x) - 2 log(4) log(x) = -log(4) log(π)
Add log^2(4) to both sides:
log^2(4) - 2 log(4) log(x) + log^2(x) = log^2(4) - log(4) log(π)
Write the left hand side as a square:
(log(x) - log(4))^2 = log^2(4) - log(4) log(π)
Take the square root of both sides:
log(x) - log(4) = sqrt(log^2(4) - log(4) log(π)) or log(x) - log(4) = -sqrt(log^2(4) - log(4) log(π))
Add log(4) to both sides:
log(x) = log(4) + sqrt(log^2(4) - log(4) log(π)) or log(x) - log(4) = -sqrt(log^2(4) - log(4) log(π))
Cancel logarithms by taking exp of both sides:
x = 4 e^sqrt(log^2(4) - log(4) log(π)) or log(x) - log(4) = -sqrt(log^2(4) - log(4) log(π))
Add log(4) to both sides:
x = 4 e^sqrt(log^2(4) - log(4) log(π)) or log(x) = log(4) - sqrt(log(4)^2 - log(4) log(π))
Cancel logarithms by taking exp of both sides:
x = 4 e^sqrt(log^2(4) - log(4) log(π))= 2.24253438346688
or x = 4 e^(-sqrt(log^2(4) - log(4) log(π)))= 7.13478469626164
2.24253438346688 x 7.13478469626164 = 16
