CONDENSE INTO THE LOG OF A SINGLE QUANTITY:

(1) Base 7
2/3 log( z - 2 )

(2) 2 ln8 + 5 ln( z - 4 )

(3) Base 4
1/2 [ log( x + 1 ) + 2 log( x - 1 ) ] + 6 logx

(1) log(7)(z-2)^(2/3)

(2) ln [64(z-4)^5]

(3) log(4)[x^6*(x-1)sqrt(x+1)]

To condense the given expressions into a single logarithm, we can use the logarithmic properties. Here are the condensed forms of the expressions:

(1) Base 7:
2/3 log( z - 2 )
= log( ( z - 2 )^(2/3) ) [using the property: log(a)^b = b log(a)]
= log( ∛( z - 2 )^2 ) [using the property: a^(p/q) = ∛(a^p)^q]
= log( ∛( z - 2 )^2 )

(2) Natural logarithm (ln):
2 ln8 + 5 ln( z - 4 )
= ln( 8^2 ) + ln( ( z - 4 )^5 ) [using the property: ln(a) + ln(b) = ln(a*b)]
= ln( 64 ) + ln( ( z - 4 )^5 )
= ln( 64 * ( z - 4 )^5 ) [using the property: ln(a) + ln(b) = ln(a*b)]
= ln( 64( z - 4 )^5 )

(3) Base 4:
1/2 [ log( x + 1 ) + 2 log( x - 1 ) ] + 6 logx
= 1/2 log( ( x + 1 ) ) + log( ( x - 1 )^2 ) + log( x^6 ) [using the property: log(a) + log(b) = log(a*b)]
= log( ( x + 1 )^(1/2) ) + log( ( x - 1 )^2 ) + log( x^6 ) [using the property: log(a)^b = b log(a)]
= log( √( x + 1 ) ) + log( ( x - 1 )^2 ) + log( x^6 ) [using the property: a^(p/q) = √(a^p)^q]
= log( √( x + 1 ) * ( x - 1 )^2 * x^6 ) [using the property: log(a) + log(b) = log(a*b)]
= log( x * √( x + 1 ) * ( x - 1 )^2 * x^5 ) [rearranging terms]
= log( x^8 * ( x + 1 )^(1/2) * ( x - 1 )^2 ) [rearranging and combining powers]

Therefore, the condensed forms are:

(1) Base 7:
log( ∛( z - 2 )^2 )

(2) Natural logarithm (ln):
ln( 64( z - 4 )^5 )

(3) Base 4:
log( x^8 * ( x + 1 )^(1/2) * ( x - 1 )^2 )

To condense each expression into the logarithm of a single quantity, we will use logarithmic properties and simplify the expressions step by step.

(1) Base 7: 2/3 log(z - 2)

We can rewrite 2/3 as a logarithmic property:
2/3 log(z - 2) = log((z - 2)^(2/3))

So the condensed form is log((z - 2)^(2/3)).

(2) 2 ln(8) + 5 ln(z - 4)

Using the logarithmic property ln(a) = log_e(a) (natural logarithm is base e),
2 ln(8) + 5 ln(z - 4) = log_e(8^2) + log_e((z - 4)^5)

Simplifying further, we have:
= log_e(64) + log_e((z - 4)^5)
= log_e(64 * (z - 4)^5)
= log_e(64z - 256)^5

So the condensed form is log_e(64z - 256)^5.

(3) Base 4: 1/2 [ log(x + 1) + 2 log(x - 1) ] + 6 log(x)

We can use logarithmic properties to simplify this expression step by step:
1/2 [ log(x + 1) + 2 log(x - 1) ] + 6 log(x)
= 1/2 log((x + 1)) + log((x - 1)^2) + log(x^6)
= log(sqrt(x + 1)) + log((x - 1)^2) + log(x^6)
= log(sqrt(x + 1) * (x - 1)^2 * x^6)

So the condensed form is log(sqrt(x + 1) * (x - 1)^2 * x^6).

Note: The base of the logarithm depends on the given problem or context. In the above explanations, I assumed base 7 for the first expression, base e (natural logarithm) for the second expression, and base 4 for the third expression.