Solve and find the domain of the equation:

\(\log_3 ((\log_{0.5}x)^2-(3\log_{0.5}x) +5)=2\)

michaelcai Feb 12, 2018

#1**+1 **

Solve for x over the real numbers:

log(3, log_0.5^2(x) - 3 log(0.5, x) + 5) = 2

log(3, log_0.5^2(x) - 3 log(0.5, x) + 5) = log(5 + 4.32809 log(x) + 2.08137 log^2(x))/log(3):

log(5 + 4.32809 log(x) + 2.08137 log^2(x))/log(3) = 2

Multiply both sides by log(3):

log(5 + 4.32809 log(x) + 2.08137 log^2(x)) = 2 log(3)

2 log(3) = log(3^2) = log(9):

log(5 + 4.32809 log(x) + 2.08137 log^2(x)) = log(9)

Cancel logarithms by taking exp of both sides:

5 + 4.32809 log(x) + 2.08137 log^2(x) = 9

Divide both sides by 2.08137:

2.40227 + 2.07944 log(x) + log^2(x) = 4.32408

Subtract 2.40227 from both sides:

2.07944 log(x) + log^2(x) = 1.92181

Add 1.08102 to both sides:

1.08102 + 2.07944 log(x) + log^2(x) = 3.00283

Write the left hand side as a square:

(log(x) + 1.03972)^2 = 3.00283

Take the square root of both sides:

log(x) + 1.03972 = 1.73287 or log(x) + 1.03972 = -1.73287

Subtract 1.03972 from both sides:

log(x) = 0.693147 or log(x) + 1.03972 = -1.73287

Cancel logarithms by taking exp of both sides:

x = 2. or log(x) + 1.03972 = -1.73287

Subtract 1.03972 from both sides:

x = 2. or log(x) = -2.77259

Cancel logarithms by taking exp of both sides:

**x = 2 or x = 0.0625**

**Domain:{x element R : x>0} (all positive real numbers) (assuming a function from reals to reals)**

Guest Feb 13, 2018