Expand the Logarithmic Expression log of (x^5)/( square root of zy)

Solution:

Step:1

Transform $log\left(\frac{x^5}{\sqrt{zy}}\right)$ into $log(x^5)-log(\sqrt{zy})$.

Step:2

Express $\sqrt{zy}$ as $(zy{)}^{\frac{1}{2}}$ using the property $\sqrt[n]{a^x}=a^{\frac{x}{n}}$.

Step:3

Apply the logarithm power rule to $log(x^5)$, bringing the exponent down: $5\cdot log(x)$.

Step:4

Similarly, apply the power rule to $log((zy{)}^{\frac{1}{2}})$, bringing $\frac{1}{2}$ down: $\frac{1}{2}\cdot log(zy)$.

Step:5

Combine the $\frac{1}{2}$ with the logarithm to get $\frac{log(zy)}{2}$.

Step:6

Split $log(zy)$ into $log(z)+log(y)$ using the logarithm product rule.

Step:7

To create a common denominator, multiply $5\cdot log(x)$ by $\frac{2}{2}$.

Step:8

Merge $5\cdot log(x)$ with $\frac{2}{2}$ to form $\frac{5\cdot log(x)\cdot 2}{2}$.

Step:9

Combine the terms over the common denominator $\frac{2}{2}$.

Step:10

Simplify the numerator.

Step:10.1

Calculate $2\cdot 5$ to get $10$.

Step:10.2

Distribute the subtraction across $log(z)$ and $log(y)$.

The final expression is $\frac{10\cdot log(x)-log(z)-log(y)}{2}$.

Knowledge Notes:

To solve the given problem, we utilize several properties of logarithms:

1. Quotient Rule: The logarithm of a quotient is the difference of the logarithms: $log(\frac{a}{b})=log(a)-log(b)$.

2. Power Rule: The logarithm of a power is the exponent times the logarithm of the base: $log(a^n)=n\cdot log(a)$.

3. Product Rule: The logarithm of a product is the sum of the logarithms: $log(ab)=log(a)+log(b)$.

4. Radical as Power: A radical can be expressed as a fractional exponent: $\sqrt[n]{a^x}=a^{\frac{x}{n}}$.

In the solution, we applied these rules step by step to expand the logarithmic expression. The process involved breaking down the expression into simpler parts, applying the rules, and then combining the terms to reach the final expanded form.