To rewrite the expression $\log \left( \frac{x^6 y^{20}}{z^{10}} \right)$ in terms of $\log(x)$, $\log(y)$, and $\log(z)$, we’ll use the logarithmic properties.

1. Quotient Rule of Logarithms: $\log \left( \frac{a}{b} \right) = \log(a) - \log(b)$
2. Product Rule of Logarithms: $\log(ab) = \log(a) + \log(b)$
3. Power Rule of Logarithms: $\log(a^n) = n \log(a)$

Using these properties, we can transform the expression step-by-step:

$\log \left( \frac{x^6 y^{20}}{z^{10}} \right) = \log \left( x^6 y^{20} \right) - \log \left( z^{10} \right)$

Applying the Product Rule on $\log \left( x^6 y^{20} \right)$:

$= \log \left( x^6 \right) + \log \left( y^{20} \right) - \log \left( z^{10} \right)$

Now, applying the Power Rule on each term:

$= 6 \log(x) + 20 \log(y) - 10 \log(z)$

Thus, we can express the expression in the desired form:

$A \log(x) + B \log(y) + C \log(z)$

where:

• $A = 6$
• $B = 20$
• $C = -10$

Would you like further details on any of the steps?

Related Questions

1. What other transformations are possible with logarithmic properties?
2. How does the logarithmic base affect the expression simplification?
3. Can this expression be simplified further if specific values are given for $x$, $y$, and $z$?
4. How would this solution change if $\log(x)$ is replaced with natural logarithm $\ln(x)$?
5. How do logarithmic rules apply to complex numbers?

Tip

Remember: The power rule allows you to bring the exponent as a multiplier in front of the logarithm, simplifying expressions with powers significantly.