Problem 16 - Algebra A Combined Function

Question

Write each difference as a single logarithm. Assume that variables represent positive numbers. $$ \log _{7}(x+9)-\log _{7}\left(x^{2}+10\right) $$

Step-by-Step Solution

Answer

\( \log_7((x+9)/(x^2+10)) \)

Step 1: Recognize the Logarithmic Identity

Understand that the difference between two logarithms can be rewritten as a single logarithm by using the property:\[ \log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right) \] where \( A \) and \( B \) are positive.

Step 2: Apply the Logarithmic Identity

Apply the identity for the given expression:\[ \log_7(x+9) - \log_7(x^2+10) = \log_7\left(\frac{x+9}{x^2+10}\right) \] This condenses the two logarithms into a single logarithmic expression.

Step 3: Simplify if Needed

Check the expression \( \frac{x+9}{x^2+10} \) for any further simplification. In this case, the expression is already in its simplest form.

Key Concepts

Difference of Logarithms

When dealing with logarithms, one useful trick to know is how to handle the difference between two logarithms. This is an important aspect because it allows us to compress the information into a single, more manageable expression. The key tool here is a specific logarithmic identity:

• \( \log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right) \)

This identity holds because logarithms represent exponents. So, when you subtract one logarithm from another with the same base, it's equivalent to dividing their respective inputs. Remember, this technique only works when the logarithms share the same base; otherwise, the identity does not apply. Furthermore, ensure that the inputs \( A \) and \( B \) are positive numbers, as logarithms of non-positive values are undefined.

Simplifying Logarithmic Expressions

In mathematics, simplifying expressions usually makes them easier to work with. Logarithmic expressions are no exception. By simplifying, we transform a more complicated form into something simpler and often more intuitive.The challenge often lies in recognizing when a logarithmic expression can be simplified. Consider the expression \( \log_7(x+9) - \log_7(x^2+10) \): it may initially seem complex, but thanks to the difference of logarithms identity, we can rewrite it as:

• \( \log_7\left(\frac{x+9}{x^2+10}\right) \)

This version is much easier to understand and manipulate. It reveals the relationship between \( x+9 \) and \( x^2+10 \) in a direct and compact form. Simplifying is all about spotting such possibilities and using logarithmic properties effectively.

Logarithmic Properties

Logarithms are governed by several foundational properties that allow us to manipulate and transform logarithmic expressions seamlessly. These properties are core tools you'll often find yourself using to simplify or combine logarithms.Some of the most essential logarithmic properties include:

• Product Property: \( \log_b(A \times C) = \log_b(A) + \log_b(C) \)
• Quotient Property: \( \log_b\left(\frac{A}{C}\right) = \log_b(A) - \log_b(C) \)
• Power Property: \( \log_b(A^C) = C \cdot \log_b(A) \)

In our context, the quotient property was especially relevant. It allows us to write the difference of two logarithms as a single logarithm, where the inputs of the original logs are divided. Understanding these properties is essential for anyone looking to master logarithmic functions and expressions. They turn what might seem like a constraint into a useful tool for tackling complex equations.