Express as a sum, difference, and product of logarithms, without using exponents

log b sqrt x^3y^2/z9

To express log base b of the square root of (x^3 * y^2 / z^9) as a sum, difference, and product of logarithms, we can use the properties of logarithms and rewrite the expression step by step.

Step 1: Start by writing the given expression:

log b √(x^3 * y^2 / z^9)

Step 2: Apply the property of square roots:

√(x^3 * y^2 / z^9) is equivalent to (x^3 * y^2 / z^9)^(1/2)

Now we have:

log b ((x^3 * y^2 / z^9)^(1/2))

Step 3: Apply the property of exponents within the logarithm:

The exponent 1/2 can be rewritten as a fraction using the property:

log b ((x^3 * y^2 / z^9)^(1/2)) = (1/2) * log b (x^3 * y^2 / z^9)

Step 4: Apply the properties of logarithms to split the logarithm of a quotient into a difference of logarithms:

(1/2) * log b (x^3 * y^2 / z^9) = (1/2) * (log b x^3 + log b y^2 - log b z^9)

Step 5: Apply the property of exponents within the logarithm:

(1/2) * (log b x^3 + log b y^2 - log b z^9) = (1/2) * (3 * log b x + 2 * log b y - 9 * log b z)

Therefore, log base b of the square root of (x^3 * y^2 / z^9) can be expressed as:

(1/2) * (3 * log b x + 2 * log b y - 9 * log b z)

So, the expression is now represented as a sum of logarithms.