Problem 49 - Algebra A Combined Function

Question

Write each expression as a sum or difference of logarithms. Assume that variables represent positive numbers. $$ \log _{5} x^{3}(x+1) $$

Step-by-Step Solution

Answer

The expression is \(3 \cdot \log_{5}(x) + \log_{5}(x+1)\).

Step 1: Apply Logarithmic Product Rule

The first step is to recognize that the expression inside the logarithm is a product: \( x^3 \cdot (x+1) \). The logarithmic property for a product states that \( \log_b(M\cdot N) = \log_b M + \log_b N \). Therefore, we can rewrite \( \log_{5}(x^3(x+1)) \) as:\[\log_{5}(x^3) + \log_{5}(x+1)\]

Step 2: Apply Logarithmic Power Rule

Next, apply the power rule of logarithms, which is \( \log_b(M^n) = n \cdot \log_b(M) \). For the term \( \log_{5}(x^3) \), we can apply this rule:\[ \log_{5}(x^3) = 3 \cdot \log_{5}(x)\]Now substitute this back into your expression:\[3 \cdot \log_{5}(x) + \log_{5}(x+1)\]

Key Concepts

Product Rule in Logarithms

The product rule in logarithms is a key identity that simplifies the logarithm of a product into the sum of two separate logarithms. If you have an expression like \( \log_b(M \cdot N) \), where you are taking the logarithm of a product, you can use the product rule to rewrite it as \( \log_b(M) + \log_b(N) \). This is incredibly useful in algebra when working with logarithmic expressions.

• Think of it as splitting the product inside the logarithm into two distinct parts.
• The base of the logarithm, \( b \), remains the same for each term.

For example, in our exercise, the expression \( x^3(x+1) \) was initially a single, more complex logarithmic expression. By applying the product rule, it became two separate terms: \( \log_{5}(x^3) + \log_{5}(x+1) \).
This rule is especially powerful as it allows for further manipulation, such as applying other logarithmic rules like the power rule.

Power Rule in Logarithms

Once you have separated the product into two terms using the product rule, the power rule in logarithms can further simplify these terms. The power rule states that if you have a logarithmic expression like \( \log_b(M^n) \), you can move the exponent \( n \) in front of the logarithm as a coefficient. This transforms the expression to \( n \cdot \log_b(M) \).

• This is useful when you have a term raised to a power within a logarithm.
• It makes dealing with exponents in logarithms a lot simpler and more straightforward.

In the example from our exercise, for the term \( \log_{5}(x^3) \), we apply the power rule, giving us \( 3 \cdot \log_{5}(x) \).
This kind of transformation is essential, as it simplifies the expression dramatically, making further calculations easier. It also allows for clear and organized answers which are simpler to interpret.

Logarithmic Expressions

A logarithmic expression is any mathematical expression that involves a logarithm. These expressions require manipulation through logarithmic identities like product and power rules to simplify or solve them.

• It is important to assume variables represent positive numbers in logarithmic expressions because the logarithm is only defined for positive values in the real number system.
• Using rules like the product and power rules helps in breaking down complex logarithmic expressions into manageable parts.

These expressions are pervasive across calculus, algebra, and beyond, forming the foundation for solving logarithmic equations and inequalities.
In our given exercise, understanding how to dissect a logarithmic expression through rules helps students efficiently write expressions as a sum or difference, which is often a requirement in solving more complex math problems. This process highlights the utility of mastering the basic identities of logarithms to tackle broader mathematical challenges.