Condensing Logarithmic Expressions: A Step-by-Step Guide

Have you ever wondered how to simplify complex logarithmic expressions? The properties of logarithms provide a powerful set of tools to condense logarithmic expressions into a more manageable form. In this guide, we'll break down the process step-by-step, focusing on how to condense the expression $7 \log_b x + 5 \log_b z$ using these valuable properties. Let's dive in and unlock the secrets of logarithmic simplification!

Understanding the Properties of Logarithms

Before we tackle the specific expression, it's crucial to understand the fundamental properties of logarithms that make this condensation possible. These properties are the building blocks for simplifying and manipulating logarithmic expressions. Let's explore the key properties that we'll be using:

• The Power Rule: This property states that $\log_b(x^p) = p \log_b(x)$. In simpler terms, when you have a logarithm of a number raised to a power, you can move the exponent in front of the logarithm as a coefficient. This is one of the most frequently used properties when condensing logarithmic expressions.
• The Product Rule: This property tells us that $\log_b(xy) = \log_b(x) + \log_b(y)$. The logarithm of a product is equal to the sum of the logarithms of the individual factors. This rule is essential for combining multiple logarithmic terms into a single logarithm.
• The Quotient Rule: This property states that $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$. The logarithm of a quotient is equal to the difference between the logarithms of the numerator and the denominator. This rule is helpful when dealing with expressions involving division within logarithms.

These three properties, the power rule, the product rule, and the quotient rule, form the basis for condensing logarithmic expressions. By understanding and applying these rules, we can transform complex expressions into simpler, more concise forms. In the following sections, we will apply these rules to our target expression and demonstrate the condensation process step-by-step.

Applying the Power Rule

Our starting expression is $7 \log_b x + 5 \log_b z$. The first step in condensing this expression involves applying the power rule. Remember, the power rule states that $\log_b(x^p) = p \log_b(x)$. This allows us to rewrite terms where a logarithm is multiplied by a coefficient.

In our expression, we have two such terms: $7 \log_b x$ and $5 \log_b z$. Applying the power rule to the first term, we can move the coefficient 7 as the exponent of $x$, resulting in $\log_b(x^7)$. Similarly, for the second term, we move the coefficient 5 as the exponent of $z$, which gives us $\log_b(z^5)$.

So, after applying the power rule, our expression now looks like this: $\log_b(x^7) + \log_b(z^5)$. We have successfully transformed the coefficients into exponents within the logarithmic terms. This is a crucial step in the condensation process, as it prepares the expression for the next stage, where we will use the product rule to combine the logarithmic terms.

By understanding and correctly applying the power rule, we have taken the first significant step towards simplifying our logarithmic expression. This rule is a fundamental tool in manipulating logarithms and is frequently used in various mathematical and scientific applications. Now, let's move on to the next step and see how the product rule can further condense our expression.

Utilizing the Product Rule

Now that we've applied the power rule and our expression is in the form $\log_b(x^7) + \log_b(z^5)$, we can move on to the next step: utilizing the product rule. The product rule, as we discussed earlier, states that $\log_b(xy) = \log_b(x) + \log_b(y)$. This rule allows us to combine the sum of two logarithms with the same base into a single logarithm of the product of their arguments.

In our current expression, we have the sum of two logarithms with the same base, $b$: $\log_b(x^7)$ and $\log_b(z^5)$. According to the product rule, we can combine these into a single logarithm by multiplying their arguments. This means we multiply $x^7$ and $z^5$.

Applying the product rule, we get: $\log_b(x^7z^5)$. We have successfully condensed the two logarithmic terms into a single term. This is the key to simplifying logarithmic expressions and making them more manageable.

By carefully applying the product rule, we have taken another significant step towards our goal of condensing the original expression. We've transformed the sum of two logarithms into a single logarithm of a product. This demonstrates the power and elegance of logarithmic properties in simplifying complex mathematical expressions. In the next section, we'll review the condensed expression and highlight the final result.

The Final Condensed Expression

After applying both the power rule and the product rule, we have successfully condensed the original expression $7 \log_b x + 5 \log_b z$ into a single logarithmic term. Let's recap the steps we took:

1. Applied the Power Rule: We moved the coefficients 7 and 5 as exponents of $x$ and $z$, respectively, resulting in $\log_b(x^7) + \log_b(z^5)$.
2. Utilized the Product Rule: We combined the two logarithms into a single logarithm by multiplying their arguments, giving us $\log_b(x^7z^5)$.

Therefore, the final condensed expression is $\log_b(x^7z^5)$. This is a much simpler and more compact representation of the original expression. It showcases the power of logarithmic properties in simplifying mathematical expressions and making them easier to work with.

By following these steps, we have demonstrated how to effectively condense logarithmic expressions using the properties of logarithms. This skill is invaluable in various areas of mathematics, science, and engineering, where logarithmic functions are frequently encountered. Understanding these properties allows us to manipulate and simplify expressions, making complex problems more manageable and understandable. This concludes our step-by-step guide on condensing the logarithmic expression $7 \log_b x + 5 \log_b z$.

Conclusion

In conclusion, condensing logarithmic expressions is a fundamental skill in mathematics, and by understanding and applying the properties of logarithms, we can simplify complex expressions into more manageable forms. We successfully condensed the expression $7 \log_b x + 5 \log_b z$ into $\log_b(x^7z^5)$ by utilizing the power rule and the product rule. Remember, the power rule allows us to move coefficients as exponents, and the product rule enables us to combine the sum of logarithms into a single logarithm of a product.

Mastering these properties opens doors to solving a wide range of mathematical problems and is crucial for anyone working with logarithmic functions. Keep practicing and applying these rules, and you'll become proficient in simplifying and manipulating logarithmic expressions. For further exploration and practice, you can visit trusted resources like Khan Academy's Logarithm Section.