Question

Condense the logarithmic expression. $$5 \ln 2+7 \ln x+4 \ln y$$

Short answer

\(\ln (32x^7y^4)\)

Step-by-step solution

Apply Power Rule

Take each coefficient and make it the power of the logarithm using the power rule of logarithms, \(a \ln b = \ln (b^a)\). This will transform the expression into: \(\ln 2^5 + \ln x^7 + \ln y^4 = \ln 32 + \ln x^7 + \ln y^4.\)

Apply Product Rule

Apply the product rule of logarithms, \(\ln a + \ln b = \ln (ab)\), to turn the sum of the logs into the log of a product. This will give us: \(\ln (32 \cdot x^7 \cdot y^4)\).

Further Simplify Expression

Simplify the expression within the logarithm: \(\ln (32x^7y^4)\).

Key concepts

Power Rule of Logarithms

The power rule of logarithms is a handy tool when dealing with logarithmic expressions. It enables us to transform multiplication within a logarithm into a simpler form. Specifically, if you have a logarithm with a coefficient, like \(a \ln b\), the power rule allows you to move that coefficient as an exponent inside the logarithm's argument. So, \(a \ln b\) becomes \(\ln (b^a)\).

This transformation is crucial in simplifying expressions, as seen in the original exercise. By changing \(5 \ln 2\) into \(\ln (2^5)\), \(7 \ln x\) into \(\ln (x^7)\), and \(4 \ln y\) into \(\ln (y^4)\), we take a big step toward condensing the logarithmic expression. Here’s what happens:

• \(5 \ln 2\) becomes \(\ln 32\).
• \(7 \ln x\) turns into \(\ln (x^7)\).
• \(4 \ln y\) changes to \(\ln (y^4)\).

This process helps clean up the expression, making it easier to combine further.

Product Rule of Logarithms

The product rule of logarithms is another fundamental concept that helps simplify expressions. It states that the sum of two logarithms is equal to the logarithm of the product of their arguments. In mathematical terms, \(\ln a + \ln b = \ln (ab)\).

Using the product rule, we can take separate logarithmic terms and combine them into a single logarithmic expression. This is exactly what happens in Step 2 of the original solution. After applying the power rule, you end up with separate terms like \(\ln 32, \ln (x^7),\) and \(\ln (y^4)\).

Here's how you condense them:

• \(\ln 32 + \ln (x^7)\) results in \(\ln (32x^7)\).
• Then, \(\ln (32x^7) + \ln (y^4)\) simplifies to \(\ln (32x^7y^4)\).

This step is essential as it combines several terms into a single, more understandable expression, where everything is contained within one logarithmic function.

Condensing Logarithms

Condensing logarithms refers to the process of taking an expanded logarithmic expression and transforming it into a single, compact form. This often involves using the power and product rules of logarithms. The goal of condensing is to create a more manageable and comprehensible expression.

In the original exercise, condensing involves combining multiple logarithmic terms like \(5 \ln 2, 7 \ln x, \) and \(4 \ln y\) into one term \(\ln (32x^7y^4)\).

The steps to condense are straightforward:

• Use the power rule to convert coefficients into exponents within the logs, simplifying each term individually.
• Apply the product rule to combine all the logarithmic terms into a single expression.

The result is a cleaner, single-expression logarithm that is easy to analyze and understand. The ability to condense logarithms is invaluable in higher mathematics, as it simplifies complex equations and facilitates easier computation.