Problem 22 - Algebra 2

Question

Expand each logarithm. \(\log 3 m^{4} n^{-2}\)

Step-by-Step Solution

Answer

\(4 \log 3m - 2 \log 3n\)

Step 1: Identify base of logarithm

The logarithm's base is 10 here, as it's not explicitly mentioned.

Step 2: Recognize and use the logarithm power rule

The power rule of logarithms states that an exponent in the argument of a logarithm can be brought to the front of the expression. Therefore, we apply it to each term inside the log separately which simplifies to \(4 \log 3m - 2 \log 3n\).

Step 3: Recognize and use the multiplication property of logarithm

When two logarithms are separated by a multiplication sign, this is equivalent to the sum of the value of the two logs. Similarly, the negative sign indicates division which is read as subtraction. So, the expression can now be rewritten as \(\log 3m^4 - \log 3n^2\), which expands to \(4 \log 3m - 2 \log 3n\).

Key Concepts

Logarithm Power Rule

The logarithm power rule is an essential concept in understanding how to simplify logarithmic expressions. This rule tells us that if you have a logarithm of a number raised to a power, you can simplify it by moving the exponent in front of the logarithm. To visualize this, consider the following:

• If you have a term like \(\log a^b\), you can rewrite it as \(b \cdot \log a\).
• This principle greatly simplifies complex logarithmic expressions and is indispensable when expanding logarithmic terms.

In our example, we can break down the expression \(\log(3 m^4 n^{-2})\) using this rule. Apply the power rule separately: the \(m^4\) becomes \(4 \log 3m\) and the \(n^{-2}\) becomes \(-2 \log 3n\). Adjust the exponents as coefficients in front of their respective logarithmic expressions, giving you the advantage of easily managing the terms.

Multiplication Property of Logarithms

The multiplication property of logarithms is another key tool for expanding and simplifying bulkier expressions. With this property, multiplying inside a logarithm can be expressed as a sum of separate logarithms. Here's how it works:

• Say you have \(\log (a \times b)\), the expression can be expanded to \(\log a + \log b\).
• Understanding this property allows you to untangle expressions into more manageable pieces.

In our exercise, this property first helps separate \(3m^4\) and \(n^{-2}\). So the step \(\log(3m^4) - \log(n^{-2})\) is simplified using the property. It must be noted that multiplication between \(m^4\) and \(n^{-2}\) within the log becomes a subtraction between them after using the property, due to \(n^{-2}\)'s exponent being negative.

Base of Logarithm

The base of a logarithm is a foundational concept in mathematical studies, though it is sometimes assumed without explicit mention. In cases where the base isn't specified, we typically assume it to be 10. Here are points to consider:

• The notation \(\log_{b} a\) indicates a logarithm with base \(b\).
• Common bases include 10, noted simply as \(\log\), and 2 or \(e\).
• Being aware of the base is crucial, especially when solving equations or simplifying expressions.

In the exercise example, the base is implicitly 10. This is typical for many textbook problems unless otherwise indicated.