To solve the first equation, we first simplify it by using the properties of logarithms.
Given: log1/2(log3(x-46)) = -Rewrite this equation using the change-of-base formula for logarithmslog(log3(x-46))/log(1/2) = -log(log3(x-46))/log(2^-1) = -log(log3(x-46))/log(2^-1) = -log(log3(x-46))/(-log2) = --log(log3(x-46))/log2 = -log3(x-46) = log2^-log3(x-46) = -x-46 = 3^-x-46 = 1/x = 46 + 1/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) (7-33.5x4-5x = 2(7-33.5x4-5x = 14 - 6762x = 1x = 10/6x = 5/31
Therefore, the solutions to the given equations are x = 46.333 and x = 5/31.
To solve the first equation, we first simplify it by using the properties of logarithms.
Given: log1/2(log3(x-46)) = -
Rewrite this equation using the change-of-base formula for logarithms
log(log3(x-46))/log(1/2) = -
log(log3(x-46))/log(2^-1) = -
log(log3(x-46))/log(2^-1) = -
log(log3(x-46))/(-log2) = -
-log(log3(x-46))/log2 = -
log3(x-46) = log2^-
log3(x-46) = -
x-46 = 3^-
x-46 = 1/
x = 46 + 1/
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 - 67
62x = 1
x = 10/6
x = 5/31
Therefore, the solutions to the given equations are x = 46.333 and x = 5/31.