If , find by logarithmic differentiation.

Question

If  , find  by logarithmic differentiation.

Quizplus · Accepted Answer

Taking the logarithmic derivative of both sides of the given expression:
$\frac{d}{dx}[(11ead0bc_8d55_2d8e_99a0_99faa7042c48_TB6988_11)^x]=\frac{d}{dx}[(11ead0bc_8d55_2d8f_99a0_051497efe003_TB6988_11)^y]$
$(11ead0bc_8d55_2d8e_99a0_99faa7042c48_TB6988_11)^x	imes\ln(11ead0bc_8d55_2d8e_99a0_99faa7042c48_TB6988_11) = (11ead0bc_8d55_2d8f_99a0_051497efe003_TB6988_11)^y	imes\ln(11ead0bc_8d55_2d8f_99a0_051497efe003_TB6988_11)$
Now substitute $x=3$ and $y=4$. We get:
$(11ead0bc_8d55_2d8e_99a0_99faa7042c48_TB6988_11)^3	imes\ln(11ead0bc_8d55_2d8e_99a0_99faa7042c48_TB6988_11) = (11ead0bc_8d55_2d8f_99a0_051497efe003_TB6988_11)^4	imes\ln(11ead0bc_8d55_2d8f_99a0_051497efe003_TB6988_11)$
Solving for $(11ead0bc_8d55_2d8f_99a0_051497efe003_TB6988_11)$, we get:
$(11ead0bc_8d55_2d8f_99a0_051497efe003_TB6988_11)=\sqrt[\leftroot{-2}\uproot{2}4]{(11ead0bc_8d55_2d8e_99a0_99faa7042c48_TB6988_11)^3	imes\ln(11ead0bc_8d55_2d8e_99a0_99faa7042c48_TB6988_11)}$
$(11ead0bc_8d55_2d8f_99a0_051497efe003_TB6988_11)=\sqrt[\leftroot{-2}\uproot{2}4]{(11ead0bc_8d55_2d8e_99a0_99faa7042c48_TB6988_11)^3}	imes\sqrt[\leftroot{-2}\uproot{2}4]{\ln(11ead0bc_8d55_2d8e_99a0_99faa7042c48_TB6988_11)}$
$(11ead0bc_8d55_2d8f_99a0_051497efe003_TB6988_11)=(11ead0bc_8d55_54a2_99a0_a793e67acdfd_TB6988_11)(\ln(11ead0bc_8d55_2d8e_99a0_99faa7042c48_TB6988_11))^{\frac{1}{4}}$
Hence the answer is option C.

If , Find by Logarithmic Differentiation