Evaluate the value for the expression (sqrt(x))^(4*log_x(a))

[suhasbadiger]

Please help me in finding the value of given question:

The value of [imath]\left( \sqrt{x} \right)^{4\, \log_x(a)}[/imath] is:

(i) [imath]a[/imath]
(ii) [imath]ax[/imath]
(iii) [imath]\frac{1}{2}\,ax[/imath]
(iv) [imath]a^2[/imath]

 

[Dr.Peterson]

Please help me in finding the value of given question in attachment

The value of [imath]\left( \sqrt{x} \right)^{4\, \log_x(a)}[/imath] is:

(i) [imath]a[/imath]
(ii) [imath]ax[/imath]
(iii) [imath]\frac{1}{2}\,ax[/imath]
(iv) [imath]a^2[/imath]

We're eager to help, but we need to see where you need help; our help doesn't consist of just giving you the answer.

Please show what you have tried, and where you are stuck. I might start, if I were you, by writing the radical as a fractional power, using exponent properties to simplify the expression, then use the inverse property of the log to finish up.

 

[Steven G]

Hint:
1) write \(\displaystyle \sqrt x\ as\ x^{1/2}\)
2) note that x^{log_x(anything)} = anything and rlog_xa = log_xa^r

 

[NotJeffM]

You are studying algebra. Use it.

[math] y = ( \sqrt{x} )^{4 \log_x (a)} = (x^{1/2})^{4 \log_x(a)} = x^{2 \log_x(a)} \implies \\ \log_x (y) = \log_x(x^{2\log_x(a)})= 2\log_x(a) * \log_x(x) = 2 \log_x(a) * 1 = 2 \log_x(a).\\ \text {Now continue and solve for y.} [/math]
This is effectively the same as the hint that Stephen G gave you. The difference is that mine relies on memorizing fewer log rules and instead applying more algebra.