Problem 66 - College Algebra and Calculus: An Applied Approach

Question

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\log _{3}\left(3^{2} \cdot 4^{2}\right)\)

Step-by-Step Solution

Answer

The expression \( \log _{3}\left(3^{2} \cdot 4^{2}\right) \) expands to \(2 + 2 \log_3 4\).

Step 1: Apply the Product Rule

The product rule states that the logarithm of a product is the sum of logarithms. It is expressed as \(\log_b (mn) = \log_b m + \log_b n\). Apply this rule to \(\log _{3}\left(3^{2} \cdot 4^{2}\right)\), which gives \(\log_3 3^2 + \log_3 4^2\). So, first the product is split into two separate logarithms.

Step 2 : Apply the Power Rule

The power rule states that the logarithm of a number to an exponent is equal to that exponent times the logarithm of the number. It is expressed as \(\log_b m^n = n \log_b m\). Apply this rule to both terms, which gives \(2 \log_3 3 + 2 \log_3 4\).

Step 3: Simplify the logarithm

Now, the expression \(2 \log_3 3 + 2 \log_3 4\) is simplified. \( \log_3 3\) is \(1\) because 3 to the power of 1 is 3 and \(\log_3 4\) will remain as is since the log base and the number are not the same. So, the final simplified expression will be \(2*1 + 2 \log_3 4\) or \(2 + 2 \log_3 4\).

Key Concepts

Product Rule

The Product Rule of logarithms is a fundamental property that helps simplify expressions where two numbers are multiplied together and under a single logarithm. If you have an expression like \(\log_b (mn)\), this can be expanded to \(\log_b m + \log_b n\). This rule is essential because it turns the multiplication inside the logarithm into an addition outside of it.

For example, in the statement \(\log _{3}\left(3^{2} \cdot 4^{2}\right)\), applying the product rule allows us to split this into \(\log_3 3^2 + \log_3 4^2\). Breaking apart the product in this way sets the stage for further simplification using other logarithmic properties. By converting multiplication into addition, calculations often become much easier to handle.

Power Rule

The Power Rule is another important tool in manipulating logarithmic expressions. This rule states that if you have a logarithm of a power, like \(\log_b m^n\), it can be written as \(n \log_b m\).

What this means is that the exponent in the expression comes out as a multiplier in front of the logarithm. In essence, it allows us to "bring down" the exponent for easier calculation. For the example \(\log_3 3^2 + \log_3 4^2\), applying the power rule converts it to \(2 \log_3 3 + 2 \log_3 4\).

• Notice that the exponent "2" in both terms simply multiplies the respective logarithmic part of the expression.
• This simplifies the expression and prepares it for further evaluation.

Understanding how the Power Rule works helps in reducing complex logarithmic expressions to more manageable forms.

Expanding Logarithmic Expressions

Expanding logarithmic expressions involves using properties like the Product and Power Rules to break down a complex logarithm into simpler parts. This method is particularly useful for both simplifying expressions and solving logarithmic equations.

The process usually involves:

• First applying the Product Rule to handle any multiplied components within the logarithm.
• Then using the Power Rule to simplify any powers by bringing the exponent in front.
• Finally, logically combining the simplified terms to produce a fully expanded expression.

In our example, starting with \(\log _{3}\left(3^{2} \cdot 4^{2}\right)\), we've seen how it expands to \(2 + 2 \log_3 4\) after a few transformations. The reason for expanding is to simplify the expression as much as possible, making it easier to understand or solve. The ability to expand logarithmic expressions is a key skill in algebra that greatly aids in the study of more advanced mathematics.