1. Write the given expression as a single logarithm.

log(70x)+log(10y)-2

2. Simplify the expression. (Do not use mixed numbers in your answer.)
(5x^7/8)(2y^4/6)(x^9/8)(4y^-2/6)

3. Write the given expression as a single logarithm.
-2Ln(x)+Ln(4y)-Ln(5z)

4. Simplify the expression. Assume that all letters represent positive real numbers. (Enter each exponent as an integer or a fraction.)
√ab ∛(ab^4)/√a(∛b)^4

another student 'anonymous' had this question,

http://www.jiskha.com/display.cgi?id=1289067581

looks very similar to yours.

1. To write the given expression as a single logarithm, we can use the rules of logarithms. The rule that will be helpful here is the addition rule, which states that log(base a) (x) + log(base a) (y) = log(base a) (xy).

So, using this rule, we can rewrite the given expression as:
log(70x) + log(10y) - 2
= log(70x * 10y) - 2
= log(700xy) - 2

Therefore, the given expression as a single logarithm is log(700xy) - 2.

2. To simplify the expression (5x^7/8)(2y^4/6)(x^9/8)(4y^-2/6), we can combine the like terms and simplify the exponents.

First, let's simplify the exponents:
(5x^7/8)(2y^4/6)(x^9/8)(4y^-2/6)
=(5 * 2 * 4 * x^(7/8 + 9/8) * y^(4/6 - 2/6))

Now, let's combine the like terms:
=(40 * x^(16/8) * y^(2/6))
=(40 * x^2 * y^(1/3))

The simplified expression is 40x^2y^(1/3).

3. To write the given expression as a single logarithm, we can use the properties of logarithms. The rule that will be helpful here is the subtraction rule, which states that Ln(x) - Ln(y) = Ln(x/y).

Using this rule, we can rewrite the given expression as:
-2Ln(x) + Ln(4y) - Ln(5z)
= Ln(x^(-2)) + Ln(4y) - Ln(5z)
= Ln((x^(-2) * (4y)) / 5z)

Therefore, the given expression as a single logarithm is Ln((x^(-2) * (4y)) / 5z).

4. To simplify the expression √ab ∛(ab^4)/√a(∛b)^4, we can simplify each term separately and then combine them.

Let's simplify each term:
√ab = (ab)^(1/2)
∛(ab^4) = (ab^4)^(1/3)
√a = a^(1/2)
(∛b)^4 = (b^(1/3))^4 = b^(4/3)

Now, let's combine the simplified terms:
√ab ∛(ab^4)/√a(∛b)^4
= (ab)^(1/2) * (ab^4)^(1/3) / (a^(1/2) * b^(4/3))
= (ab)^(1/2 + 1/3) / (a^(1/2) * b^(4/3))
= (ab)^(3/6 + 2/6) / (a^(1/2) * b^(4/3))
= (ab)^(5/6) / (a^(1/2) * b^(4/3))
= (ab)^(5/6 - 3/6) / a^(1/2)
= (ab)^(2/6) / a^(1/2)
= (ab)^(1/3) / a^(1/2)

Therefore, the simplified expression is (ab)^(1/3) / a^(1/2).