Expand the Logarithmic Expression log of (x^5)/( square root of zy)
The question is asking for the expansion of a given logarithmic expression using the properties of logarithms. Specifically, the expression to be expanded is the natural logarithm (log, which is often assumed to be the natural log unless otherwise specified, ln for natural logarithm) of a fraction where the numerator is x raised to the fifth power (x^5) and the denominator is the square root of the product of z and y (√(zy)). The task is to rewrite the expression in a way that separates the log of the numerator, the log of the denominator, and incorporates the exponents and roots into a more simplified, equivalent form using logarithmic identities such as the quotient rule, the power rule, and the product rule for logarithms.
$log \left(\right. \frac{x^{5}}{\sqrt{z y}} \left.\right)$
Answer
Solution:
Step:1
Transform $log \left(\frac{x^{5}}{\sqrt{z y}}\right)$ into $log(x^{5}) - log(\sqrt{z y})$.
Step:2
Express $\sqrt{z y}$ 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:
Quotient Rule: The logarithm of a quotient is the difference of the logarithms: $log(\frac{a}{b}) = log(a) - log(b)$.
Power Rule: The logarithm of a power is the exponent times the logarithm of the base: $log(a^{n}) = n \cdot log(a)$.
Product Rule: The logarithm of a product is the sum of the logarithms: $log(ab) = log(a) + log(b)$.
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.
