1. Solve log(4/x) (x^2 - 6) = 2. (note: the (4/x) is underscored)

2. Show that loga [(x + {sqrtx^2-5}) / 5] = -loga (x - (sqrtx^2-5))

kinda weird, but if you've written it correctly,

log(4/x) (x^2 - 6) = 2
raise 4/x to both sides and you have

x^2-6 = (4/x)^2
x^2 - 6 = 16/x^2
x^4 - 6x^2 - 16 = 0
(x^2+2)(x^2-8) = 0
x = ±√2 i or ±2√2

(x+√(x^2-5))/5 =

(x+√(x^2-5))(x-√(x^2-5))
---------------------
5(x-√(x^2-5))

x^2 - (x^2-5)
----------------
5(x-√(x^2-5))

1/(x-√(x^2-5))

So, log of one is -log of its reciprocal.

To solve these logarithmic equations, we need to use logarithmic properties and algebraic techniques. Let's break down each equation step by step:

1. Solve log(4/x) (x^2 - 6) = 2:

Step 1: Apply the power rule of logarithms. Using this rule, we can move the exponent in front of the logarithm:
(x^2 - 6) log(4/x) = 2

Step 2: Rewrite the logarithm using the quotient rule of logarithms:
log(4/x) - log(x) (x^2 - 6) = 2

Step 3: Apply the product rule of logarithms to simplify the equation:
log(4/x) - [(log(x))(x^2 - 6)] = 2

Step 4: Expand the expression in square brackets:
log(4/x) - (x^2 - 6)(log(x)) = 2

Step 5: Distribute the logarithm inside the parentheses:
log(4/x) - (x^2)(log(x)) + 6(log(x)) = 2

Step 6: Combine the terms with the same logarithm:
log(4/x) - x^2(log(x)) + 6(log(x)) = 2

Step 7: Move the constant term to the right side of the equation:
log(4/x) - x^2(log(x)) + 6(log(x)) - 2 = 0

Now, you may use numerical approximation methods or software/calculators capable of solving nonlinear equations to find the value(s) of x that satisfy this equation.

2. Show that loga [(x + sqrt(x^2 - 5)) / 5] = -loga (x - sqrt(x^2 - 5)):

To prove this equation, we need to apply logarithmic properties and simplifications:

Step 1: Apply the quotient rule of logarithms on the left side:
loga [(x + sqrt(x^2 - 5)) / 5] = loga(x + sqrt(x^2 - 5)) - loga(5)

Step 2: Use the power rule of logarithms to rewrite the first term on the right side:
loga(x + sqrt(x^2 - 5)) - loga(5) = loga(x) + loga(sqrt(x^2 - 5)) - loga(5)

Step 3: Simplify the logarithms using logarithmic properties:
loga(x) + loga(sqrt(x^2 - 5)) - loga(5) = loga(x) + (1/2)loga(x^2 - 5) - loga(5)

Step 4: Apply the quotient rule of logarithms on the right side:
loga(x) + (1/2)loga(x^2 - 5) - loga(5) = loga(x) - loga(x^2 - 5) - loga(5)

Step 5: Combine the last two logarithms using the difference rule of logarithms:
loga(x) - loga(x^2 - 5) - loga(5) = loga(x/(x^2 - 5)) - loga(5)

Step 6: Apply the quotient rule of logarithms to the final expression:
loga(x/(x^2 - 5)) - loga(5) = loga[(x/(x^2 - 5))/5]

Step 7: Simplify the right side by multiplying the numerator by (1/5):
loga[(x/(x^2 - 5))/5] = loga(x/(5(x^2 - 5)))

Now, both sides of the equation are in the same form. Therefore, we have successfully shown that loga[(x + sqrt(x^2 - 5)) / 5] = -loga(x - sqrt(x^2 - 5)).