Using logarithms to rewrite the expression:

1.) log4.(4)^3x=
2.) 6 log6 36=
3.) 3 log 2 1/2=
4.) 1/4 log4 16

To rewrite these expressions using logarithms, we need to understand and apply the properties of logarithms. The properties we will use are as follows:

1. The exponent property: log base b (x^n) = n log base b (x)
2. The product property: log base b (xy) = log base b (x) + log base b (y)
3. The quotient property: log base b (x/y) = log base b (x) - log base b (y)

Now, let's rewrite the given expressions using logarithms:

1.) log4 (4)^3x
Using the exponent property, we can rewrite (4)^3x as 4^(3x). Therefore, the expression becomes:
log4 (4^(3x))

2.) 6 log6 36
Using the exponent property, we can rewrite 36 as 6^2. Therefore, the expression becomes:
6 log6 (6^2)

3.) 3 log2 (1/2)
Using the quotient property, we can rewrite 1/2 as 2^(-1). Therefore, the expression becomes:
3 log2 (2^(-1))

4.) 1/4 log4 16
Using the exponent property, we can rewrite 16 as 4^2. Therefore, the expression becomes:
1/4 log4 (4^2)

By following these steps, we have rewritten the given expressions using logarithms.

by definition, log_b(b^z) = z

So,

log_4(4^3x) = 3x
6 log_6(36) = 6 log_6(6^2) = 6*2 = 12
3 log_2(1/2) = 3 log_2(2^-1)) = 3*-1 = -3
1/4 log_4(16) = 1/4 log_4(4^2) = 1/4 * 2 = 1/2