Simplify $20 \log _6 x+5 \log _6 y$

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01-11-2026
Mathematics

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Simplify $20 \log _6 x+5 \log _6 y$

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-01T08:55:52-05:00

Use the power rule of logarithms $\log_b(a^n) = n \log_b(a)$ to rewrite the terms as $\log_6(x^{20})$ and $\log_6(y^5)$.
Use the product rule of logarithms $\log_b(M) + \log_b(N) = \log_b(MN)$ to combine the terms.
The expression simplifies to $\boxed{\log_6(x^{20}y^5)}$.

Explanation

Analyzing the Expression

The given expression is $20 \log_6 x + 5 \log_6 y$. To simplify this, we must recognize the properties of logarithms. The variables $x$ and $y$ must be positive, as the logarithm function $\log_b(a)$ is defined only for $a > 0$. Our objective is to combine these two logarithmic terms into a single expression of the form $\log_6(Z)$.

Applying the Power Rule of Logarithms

We start by applying the power rule of logarithms, which states that for any real number $n$, $n \log_b a = \log_b (a^n)$. Applying this rule to each term individually:

For the first term: $20 \log_6 x = \log_6 (x^{20})$
For the second term: $5 \log_6 y = \log_6 (y^5)$

The expression now becomes $\log_6 (x^{20}) + \log_6 (y^5)$.

Combining the Terms Using the Product Rule

Next, we use the product rule of logarithms, which states that $\log_b M + \log_b N = \log_b (MN)$. Using this rule to combine our two terms:
$\log_6 (x^{20}) + \log_6 (y^5) = \log_6 (x^{20} \cdot y^5)$.
This gives us a single logarithm as required.

Final Conclusion

The expression has been successfully simplified into a single logarithmic term with base 6. The final simplified form is $\log_6 (x^{20} y^5)$. Hence, the result is $\boxed{\log_6 (x^{20} y^5)}$.

Examples

Logarithms are widely used in science and engineering to represent processes that occur across vastly different scales, such as the Richter scale for earthquake magnitude or the pH scale for acidity. By condensing complex logarithmic expressions into single terms using rules like the power and product rules, scientists can simplify mathematical models and more easily perform calculations or analyze relationships between variables, such as comparing the energy output of two different seismic events or the concentration of ions in chemical solutions. Mastering these rules is essential for manipulating these types of data-driven models.