Problem 16 - Precalculus with Limits

Question

Using Properties of Logarithms In Exercises \(15-20\) , use the properties of logarithms to rewrite and simplify the logarithmic expression. $$\log _{2}\left(4^{2} \cdot 3^{4}\right)$$

Step-by-Step Solution

Answer

The simplified form of the given logarithmic expression is 10.

Step 1: Simplify the inside of the log

Evaluate the expression inside the log. Given \(4^{2} \cdot 3^{4}\), we have to execute the power operations first. So, \(4^{2} = 16\) and \(3^{4} = 81\). The product of 16 and 81 is 1296, which results in \(\log _{2}(1296)\).

Step 2: Apply the properties of logarithms

Now, we apply the properties of logarithms to simplify the expression. We can use the fact that the logarithm base 2 of a number is the power to which 2 needs to be raised to get that number, so \(\log _{2}(2^n) = n\). We find such a power n that \(2^n = 1296\). After quick calculations, we find this power to be n = 10. So our expression simplifies to \(\log _{2}(2^{10})\). Remember that this simplification is possible because 1296 is a power of 2.

Step 3: Use the property of exponents and simplify

Finally, we can use another rule of logarithms: \(\log_b(b^a) = a\). According to this rule, our expression simplifies to 10, because \(\log _{2}(2^{10}) = 10\). In logarithms, if the base in the log is the same as the base in the exponent, they cancel out and you're left with the exponent.

Key Concepts

Logarithmic Simplification

When tasked with simplifying a logarithmic expression like \(\log _{2}(4^{2} \cdot 3^{4})\), it's crucial to take a series of organized steps. The process of logarithmic simplification involves reducing complex expressions to their simplest form. This often requires evaluating powers, products, and applying specific logarithmic properties.

For our given expression, begin by calculating the power operations within the logarithm. So, compute \(4^2\) which equals 16, and \(3^4\) which results in 81. This transforms the expression inside the logarithm to \(16 \cdot 81\), resulting in 1296, hence simplifying the logarithm to \(\log _{2}(1296)\).

Logarithmic simplification also means recognizing numbers as powers of the base whenever possible. Here, 1296 is indeed a power of 2, specifically \(2^{10}\). Using this powerful insight drastically simplifies the calculation process.

Logarithmic Expressions

Logarithmic expressions, such as \(\log_{2}(4^{2} \cdot 3^{4})\), often appear daunting at first. However, these expressions essentially describe the operation needed to raise one number (the base) to a power that equals another number. Understanding the basic makeup of logarithms can make their manipulation much more manageable.

In any expression \(\log_{b}(a)\), we're essentially asking: "To what power must \(b\) be raised, to obtain \(a\)?" This fundamental question palpably guides the transformation of complex expressions.

Using logarithmic properties, such as the product property and power property, enables us to deconstruct and reconstruct these expressions effectively.

• Product Property: \(\log_{b}(mn) = \log_{b}(m) + \log_{b}(n)\)
• Power Property: \(\log_{b}(m^n) = n \cdot \log_{b}(m)\)

By reinterpreting these expressions through their properties, complex logarithmic terms are skillfully unraveled.

Exponent Properties in Logarithms

One of the core pillars in simplifying logarithms is the understanding of exponent properties within logarithmic contexts. Here we utilize the fact that logarithms and exponents are inverse functions of each other. This relationship becomes especially useful when tackling expressions like \(\log_{2}(2^{10})\).

The most efficient way to simplify is by recognizing the property: \(\log_b(b^a) = a\). This principle directly links the base in the exponent to the base in the logarithm, simplifying the problem considerably.

The expression \(\log _{2}(2^{10})\) can be meticulously broken down using this property:

• Since the base of the logarithm (2) matches the base of the exponent (2), you can directly cancel these out.
• You're left with simply the exponent, which is 10.

This property not only speeds up calculations but also enhances our comprehension of how powerful these transformations can be in simplifying problems. Keeping in mind this trick, the task of simplifying logarithmic expressions becomes significantly more straightforward.