Question 1
Solve for x in the equation
(2x)^(lg2) = (7x)^(lg7)
Answer
(2x)^(lg2) = (7x)^(lg7)
(2^lg2)(x^lg2)=(7^lg7)(x^lg7)
(x^lg2)/(x^lg7)=(7^lg7)/(2^lg2)
x^(lg2-lg7)=(7^lg7)/(2^lg2)
Bring over the power to the other side of the equation
x = [(7^lg7)/(2^lg2)]^{1/(lg2-lg7)}
Alternative Solution
(2x)^(lg2) = (7x)^(lg7)
Taking lg on both side,
lg2(lg2+lgx) = lg7(lg7+lgx)
lgx(lg2-lg7)=(lg7)^2-(lg2)^2
lgx(lg2-lg7)=(lg7-lg2)(lg7+lg2)
lgx=-(lg7+lg2)
x=1/14
Question 2
Solve (log a X)^(log b X) = x where a, b are positive real numbers except 1, leave the answer in terms of a and b.
Answer
(loga x)^(logb x) = x
Let x = b^(logb x)
(loga x)^(logb x) = b^(logb x)
Same power, equate base
loga x = b
x = a^b
The number in red is in subscript ie the base of the log].