Log1/2log3(x-46)=-1 Log9(4-5x)+1=log9 2+log9(7-33,5x)

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To solve the first equation, we first simplify it by using the properties of logarithms.

Given: log1/2(log3(x-46)) = -1
Rewrite this equation using the change-of-base formula for logarithms:
log(log3(x-46))/log(1/2) = -1
log(log3(x-46))/log(2^-1) = -1
log(log3(x-46))/log(2^-1) = -1
log(log3(x-46))/(-log2) = -1
-log(log3(x-46))/log2 = -1
log3(x-46) = log2^-1
log3(x-46) = -1
x-46 = 3^-1
x-46 = 1/3
x = 46 + 1/3
x = 46 1/3 or 46.333

Now, let's simplify the second equation:

Given: log9(4-5x) + 1 = log9(2) + log9(7-33.5x)

Using the properties of logarithms, we can simplify this equation:

log9((4-5x)9) = log9(2) (7-33.5x)
4-5x = 2(7-33.5x)
4-5x = 14 - 67x
62x = 10
x = 10/62
x = 5/31

Therefore, the solutions to the given equations are x = 46.333 and x = 5/31.