Express as a sum, difference, and product of logarithms, without using exponents.logb^3 sqrt x^8 divided by (y^2)(z^5)

Logb^3*sqrtX^8/(Y^2)(Z^5)=

Logb^3*sqrtX^8-Log(Y^2)(Z^5)=
Logb^3+LogsqrtX^8-LogY^2+LogZ^5.

To express the given expression as a sum, difference, and product of logarithms, let's break it down step by step.

1. Start with the given expression:
logᵇ(3√x⁸ / (y²)(z⁵))

2. Using logarithm properties, we can rewrite the expression as:
logᵇ(3√x⁸) - logᵇ((y²)(z⁵))

3. Rewrite the first part of the expression as a product of logarithms:
logᵇ(3) + logᵇ(√x⁸) - logᵇ((y²)(z⁵))
(Using the logarithm property: log(AB) = log(A) + log(B))

4. Simplify the second logarithm using the power rule:
logᵇ(3) + logᵇ(x⁴) - logᵇ((y²)(z⁵))
(√x⁸ simplifies to x⁴)

5. Finally, combine the last two logarithms using the quotient rule:
logᵇ(3) + logᵇ(x⁴) - (logᵇ(y²) + logᵇ(z⁵))
(Using the logarithm property: log(A/B) = log(A) - log(B))

So, the expression logᵇ(3√x⁸ / (y²)(z⁵)) can be expressed as the sum, difference, and product of logarithms as:
logᵇ(3) + logᵇ(x⁴) - logᵇ(y²) - logᵇ(z⁵)