Problem 51 - Precalculus Mathematics for Calculus

Question

Use the Laws of Logarithms to combine the expression. $$\ln 5+2 \ln x+3 \ln \left(x^{2}+5\right)$$

Step-by-Step Solution

Answer

\( \ln(5x^2(x^2 + 5)^3) \).

Step 1: Apply the Power Rule

The Power Rule for logarithms states that \( k \ln a = \ln a^k \). Use this to rewrite \( 2 \ln x \) and \( 3 \ln (x^2 + 5) \) as \( \ln x^2 \) and \( \ln (x^2 + 5)^3 \) respectively. This gives us the expression:\[ \ln 5 + \ln x^2 + \ln (x^2 + 5)^3 \]

Step 2: Apply the Product Rule

The Product Rule for logarithms states that \( \ln a + \ln b = \ln (a \times b) \). Use this rule to combine all the logarithmic terms:First, combine \( \ln x^2 \) and \( \ln (x^2 + 5)^3 \) into a single logarithm: \( \ln (x^2 \times (x^2 + 5)^3) \).Next, include \( \ln 5 \) in the product: \( \ln (5 \times x^2 \times (x^2 + 5)^3) \).This results in the expression:\[ \ln (5x^2(x^2 + 5)^3) \]

Key Concepts

Power Rule for Logarithms

The Power Rule for Logarithms is a fundamental property that simplifies expressions where a logarithm is multiplied by a coefficient. This rule states that if you have a logarithm in the form of \( k \ln a \), you can rewrite this as \( \ln a^k \). This means you can "pull up" the coefficient to become an exponent of the term inside the logarithm.
For example, if you apply the power rule to \( 2 \ln x \), you can rewrite it as \( \ln x^2 \), transforming the coefficient into an exponent. Similarly, applying it to \( 3 \ln (x^2 + 5) \) gives \( \ln (x^2 + 5)^3 \).
This rule greatly simplifies complex logarithmic expressions by reducing the number of separate terms you have to deal with. It's especially useful in equations where you need to combine logs further, like when applying the product rule.

Product Rule for Logarithms

The Product Rule for Logarithms helps you combine multiple logarithms into one. It states that \( \ln a + \ln b = \ln (a \times b) \). This rule is handy for reducing the number of logarithmic terms and simplifying the overall expression.
When you have a logarithmic expression such as \( \ln x^2 + \ln (x^2 + 5)^3 \), you can apply the product rule to combine them into a single logarithm: \( \ln (x^2 \times (x^2 + 5)^3) \).
Following this, if you have another term like \( \ln 5 \) to add into the expression, you can continue applying the product rule: \( \ln 5 + \ln x^2 + \ln (x^2 + 5)^3 \) becomes \( \ln (5 \times x^2 \times (x^2 + 5)^3) \).
This rule simplifies complex expressions into a cleaner, single logarithm, making it easier to interpret and calculate outcomes for logarithmic problems.

Logarithmic Expressions

Logarithmic Expressions are mathematical phrases containing logarithms. These expressions can seem complex, but understanding the properties of logarithms allows you to simplify and break them down.
For instance, consider the expression \( \ln 5 + 2 \ln x + 3 \ln (x^2 + 5) \). By applying the Power Rule, the expression becomes \( \ln 5 + \ln x^2 + \ln (x^2 + 5)^3 \). Then, use the Product Rule to further simplify it into \( \ln (5x^2(x^2 + 5)^3) \).
Logarithmic expressions demand a mastery of rules like the Power Rule and the Product Rule, which allow transformations into more manageable forms. This way, you can not only simplify calculations but also gain a deeper understanding of how logarithms interact in algebraic expressions.