Problem 64 - Intermediate Algebra

Question

Write each logarithm as the sum and/or difference of logarithms of a single quantity. Then simplify, if possible. See Example 6. $$ \log x y^{2} z^{3} $$

Step-by-Step Solution

Answer

\( \log x + 2\log y + 3\log z \)

Step 1: Applying Logarithm Properties

To begin, we apply the logarithm property: \( \log(ab) = \log a + \log b \). Using this, we can express \( \log(x y^2 z^3) \) as the sum of logarithms: \( \log x + \log(y^2) + \log(z^3) \).

Step 2: Simplifying Exponents within Logarithms

Next, we will simplify each term with an exponent further using the property \( \log(a^b) = b \log a \). Applying this, we get: \( \log x + 2\log y + 3\log z \).

Step 3: Combining and Simplifying the Expression

Now, we simply combine the terms to get the final expression: \( \log x + 2 \log y + 3 \log z \). This expression cannot be simplified further as it's already expressed as a sum of logarithms.

Key Concepts

Logarithm Properties

Logarithm properties are essential tools in simplifying and transforming expressions involving logarithms. One of the most frequently used properties is the product rule, which states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, this is expressed as:

• \( \log(ab) = \log a + \log b \)

This property allows us to break down complex logarithmic expressions into simpler components that are easier to manage.
Another important property is the power rule, which helps in dealing with exponents within logarithms. According to this rule, the logarithm of a number raised to an exponent can be rewritten as the product of the exponent and the logarithm of the base:

• \( \log(a^b) = b \log a \)

In the given exercise, we can see these properties at work as they help simplify the expression \( \log(x y^2 z^3) \) into manageable parts.

Exponent Rules

Exponent rules simplify expressions involving powers, often making calculations more straightforward. One primary exponent rule is that any number raised to a power can be shifted to a logarithm's coefficient. This is handy when dealing with logarithmic expressions with power terms.
For example, given an expression such as \( y^2 \) or \( z^3 \), the exponent rules tell us that these terms can be expressed as:

• \( y^2 \rightarrow 2\log y \)
• \( z^3 \rightarrow 3\log z \)

Using exponent rules in logarithmic expressions simplifies them by turning complex multiplications into simpler additions. This transformation is particularly useful in expressions that need to be used in algebraic operations, making calculations less cumbersome and results easier to interpret.
By applying these rules, expressions such as \( \log(y^2) \) become \( 2\log y \), and \( \log(z^3) \) becomes \( 3\log z \), reducing the burdens of direct computation.

Simplifying Logarithms

Simplifying logarithmic expressions involves using both logarithm properties and exponent rules efficiently. The aim is to convert complex logarithmic expressions into a form where each component is individually understandable and easily accessible to use in further calculations.
The initial step in simplification often involves expressing a logarithm of a product as a sum using the properties of logarithms. For example, an expression like \( \log(x y^2 z^3) \) is decomposed into:

• \( \log x + \log(y^2) + \log(z^3) \)

Subsequently, exponent rules are applied, reducing each power term within a logarithm to a linear form:

• \( \log(y^2) = 2\log y \)
• \( \log(z^3) = 3\log z \)

Thus, the simplified version becomes \( \log x + 2\log y + 3\log z \).
Simplifying logarithms helps clarify each element's contribution to the overall expression, making it more manageable for further mathematical operations or solving equations.