Problem 69 - 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 \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \log x+\log \left(x^{2}-1\right)-\log 7-\log (x+1) $$

Step-by-Step Solution

Answer

The given expression simplifies to \( log((x*(x^{2}-1))/(7*(x+1))) \)

Step 1: Combine addition operation in the expression

According to the properties of logarithms, the sum of logarithms can be rewritten as the logarithm of a product. So, apply this property on the expression and combine \(\log x\) and \(\log \left(x^{2}-1\right)\) to get: \(log(x*(x^{2}-1))\)

Step 2: Combine all terms in the expression

Following the same property as in step 1, the subtraction of logarithms can be rewritten as the logarithm of a division. So, combine \(\log 7\) and \(\log (x+1)\) with the result from the previous step to get: \( log((x*(x^{2}-1))/(7*(x+1))) \)

Step 3: Simplify the expression

The current expression equals to \( log((x*(x^{2}-1))/(7*(x+1))) \). But the exercise needs the coefficient of the logarithm to be 1. Since it's already 1, there is no need for further simplification.

Key Concepts

Logarithm Simplification

Logarithm simplification is a fundamental skill in mathematics, particularly when dealing with expressions involving multiple logarithms with different bases or arguments. It involves using properties of logarithms to rewrite a series of logarithmic terms in a condensed and often more manageable form. One key property used in simplification is that the sum of logarithms with the same base can be expressed as the logarithm of the product of their arguments, symbolically represented as \( \log_b(m) + \log_b(n) = \log_b(m \cdot n) \). Conversely, if we have a difference of logarithms, such as \( \log_b(m) - \log_b(n) \), it simplifies to the logarithm of the quotient \( \log_b\left(\frac{m}{n}\right) \).

Consider the expression from the exercise \( \log x + \log(x^2 - 1) - \log 7 - \log(x + 1) \). The initial step merges the addition of logarithms, converting them into a multiplication within a single logarithmic expression, and the subtraction of logarithms becomes a division. This blending of multiple steps into one operation streamlines the process and avoids potential errors that could arise from handling many separate logarithmic terms. Simplifying logarithmic expressions in this way not only makes them easier to work with but also readies them for evaluation, either by hand or with a calculator.

Logarithmic Expressions

Logarithmic expressions are representations of the logarithm function that have various applications ranging from solving equations to modeling real-world phenomena. These expressions state that if \( \log_b(m) = c \), where \( b \), \( m \), and \( c \), are real numbers with \( b > 0 \), \( b eq 1 \), and \( m > 0 \), then \( b^c = m \). It's a way of asking for which exponent \( c \), the base \( b \), when raised to it, will produce \( m \).

Understanding and manipulating these expressions is crucial in algebra and calculus. For example, in the exercise provided, being comfortable with converting the addition or subtraction of logarithms into a single logarithmic expression is critical for condensing and solving the problem. When the expressions are combined according to the logarithmic properties, we can often find simpler forms or even explicit numerical values. Moreover, recognizing how to transition between exponential and logarithmic forms empowers students with the ability to approach and solve a wider array of mathematical problems.

Combining Logarithms

Combining logarithms is a technique derived from properties of logarithms aimed at condensing complex logarithmic expressions into a single term. This approach simplifies the expressions and makes the algebraic manipulation of logarithmic equations much more straightforward. The properties that are often invoked include:

• Product Rule: The sum of two logs with the same base is equivalent to the logarithm of the product of their arguments.
• Quotient Rule: The difference between two logs with the same base equals the logarithm of the quotient of their arguments.
• Power Rule: A logarithm with an exponent can be simplified by multiplying the exponent with the logarithm.

As seen in the solution steps of the exercise, these properties are utilized to combine four separate logarithmic terms into a single term. Such a capability is not just an abstract mathematical operation but frequently appears in various scientific fields, including physics, computer science, and engineering. After combining, the expression is neatly packaged into one logarithmic statement, which can then be more easily evaluated or further manipulated as required by the context of a problem.