Problem 59 - College Algebra

Question

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 4 \ln (x+6)-3 \ln x $$

Step-by-Step Solution

Answer

The answer is \(\ln\left(\frac{(x + 6)^4}{x^3}\right)\)

Step 1: Apply the coefficient rule

Using the rule of coefficients in logarithms, rewrite each term. The expression becomes \(\ln (x + 6)^4 - \ln x^3\)

Step 2: Apply the subtraction rule

Now apply the rule of subtraction in logarithms, which states that the difference of two logs equals the log of the quotient of the constituents. This makes the expression \(\ln\left(\frac{(x + 6)^4}{x^3}\right)\)

Step 3: Simplify the expression

You can simplify the expression within the log. Knowing that when possible, we evaluate logarithmic expressions, it is clear that in this case the final expression cannot be simplified further numerically because the variables remain.

Key Concepts

Logarithmic Expressions

Logarithmic expressions are mathematical statements that involve logarithms, which are the inverses of exponential functions. Understanding how to manipulate these expressions is crucial in algebra, calculus, and various fields of science.

A basic form of a logarithm is given by the expression \( \text{log}_b(x) \), which answers the question: 'To what exponent must we raise the base \( b \) to obtain \( x \)?' In the exercise provided, we see logarithms with the natural base, denoted by the symbol \( \text{ln} \), which is short for the natural logarithm and implies a base of \( e \), the Euler's number, approximately equal to 2.71828.

The beauty of logarithms is that they transform the multiplication and division of numbers into addition and subtraction, respectively. This makes complex exponential operations much simpler to handle. For instance, when faced with the task of condensing logarithmic expressions, students can apply a series of rules to simplifying them, such as the coefficient and subtraction rules for logarithms.

Coefficient Rule Log

The coefficient rule, sometimes known as the power rule for logarithms, is an essential property that greatly simplifies the process of working with logarithmic expressions. It states that a coefficient in front of a logarithm can be moved into the argument of the logarithm as an exponent. Mathematically, the rule is \( a \text{log}(x) = \text{log}(x^a) \).

In the exercise example, the coefficient rule is applied in the first step. The expression \( 4 \text{ln }(x+6) \) becomes \( \text{ln }((x+6)^4) \). By utilizing this rule, we're able to rewrite the original expression in a form that prepares us to combine logarithmic terms more easily. This is why understanding and correctly applying the coefficient rule is key for students when they work to condense logarithms into a single term.

Subtraction Rule Log

Following the coefficient rule, the subtraction rule log is another fundamental property used to combine logarithmic expressions. This rule tells us that the subtraction of two logarithms with the same base is the logarithm of the quotient of their arguments. Symbolically, it's expressed as \( \text{log}_b(x) - \text{log}_b(y) = \text{log}_b(\frac{x}{y}) \).

In our given exercise, after applying the coefficient rule, we encounter a subtraction of two logs. By invoking the subtraction rule, we change the expression from \( \text{ln }(x + 6)^4 - \text{ln } x^3 \) to \( \text{ln }\left(\frac{(x + 6)^4}{x^3}\right) \). This single logarithmic expression is much more concise and is closer to the solution form requested in the exercise that is a single logarithm with a coefficient of 1. Mastering the subtraction rule allows students to condense multiple logarithmic terms into one, simplifying their work and paving the way for further evaluation or solving of logarithmic equations.