evaluate each log.

Ln e^3

logbase(1/1000)

expand each log.

logbase5 radical41/43

logbase7 8xy^4/3z^3

logbase5 4(7radicalxy^z)

To evaluate each logarithm and expand each expression, we will use the properties of logarithms.

1. Evaluating ln(e^3):
The natural logarithm function (ln) is the logarithm to the base e. Since e raised to any power equals e, ln(e) = 1. Therefore, evaluating ln(e^3) simplifies to 3.

2. Evaluating logbase(1/1000):
To evaluate this logarithm, we need to know the base. You mentioned "logbase," but you did not specify the base. Please provide the base for this logarithm.

3. Expanding logbase5 radical(41/43):
Expanding the logarithm can be done by using the properties of logarithms. We can rewrite the radical as an exponent. The expression becomes logbase5 (41/43)^(1/2). Using the property logbaseb(x^a) = a * logbaseb(x), we have: (1/2) * logbase5 (41/43).

4. Expanding logbase7 (8xy^4/3z^3):
Again, we use the properties of logarithms to expand this expression. Using the property logbaseb(xy) = logbaseb(x) + logbaseb(y) and logbaseb(x^a) = a * logbaseb(x), we can rewrite the expression as logbase7(8) + logbase7(x) + (4/3) * logbase7(y) - 3 * logbase7(z).

5. Expanding logbase5 4(7 radical(xy^z)):
This expression involves a product within the log. Using the property logbaseb(xy) = logbaseb(x) + logbaseb(y), we can rewrite the expression as logbase5(4) + logbase5(7) + logbase5(radical(xy^z)). Since the radical is in the form of an exponent, we can rewrite it as logbase5(4) + logbase5(7) + logbase5((xy^z)^(1/2)). Now, using the property logbaseb(x^a) = a * logbaseb(x), we have logbase5(4) + logbase5(7) + (1/2) * logbase5(xy^z).

Remember to substitute the specific values of x, y, and z into the expanded logarithms if you have them.