Problem 72 - A Graphical Approach to College Algebra

Question

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{6}(7 m+3 q)$$

Step-by-Step Solution

Answer

\(\log_{6}(7m + 3q)\) cannot be simplified further.

Step 1: Identify the Expression

The given expression is \( \log_{6}(7m + 3q) \). This is a logarithm of a sum.

Step 2: Understand Logarithm Properties

Recall that logarithm properties such as the product rule \( \log_{b}(xy) = \log_{b}(x) + \log_{b}(y) \) and the quotient rule \( \log_{b}\left(\frac{x}{y}\right) = \log_{b}(x) - \log_{b}(y) \) do not apply to the sum or difference inside a logarithm.

Step 3: Conclusion

Since the sum form \( \log_{6}(7m + 3q) \) cannot be simplified using the standard logarithm properties, we conclude that it is already in its simplest form.

Key Concepts

Logarithms

Logarithms are mathematical expressions that help us solve equations involving exponents. They are essentially the inverse operation of exponentiation. For example, if we have a known base and exponential result, we can use logarithms to find the missing exponent.

The expression \( \log_{b}(a) \) represents the logarithm with base \( b \) of \( a \). It answers the question: "To what power must \( b \) be raised, to obtain \( a \)?" So, if you see \( \log_{2}(8) \), it asks "2 raised to what power equals 8?" The answer is 3, because \( 2^3 = 8 \).

• Natural logarithms have the base \( e \), where \( e \approx 2.718 \).
• Common logarithms have the base 10. You usually see this written as \( \log(\text{number}) \) without the base indicated.

Logarithm Properties

Logarithm properties are rules that allow us to manipulate logarithmic expressions. These properties make it easier to simplify and solve more complex equations. It's important to note which properties apply to different operations within logarithmic expressions.

Some of the most important properties include:

• Product Rule: \( \log_{b}(xy) = \log_{b}(x) + \log_{b}(y) \). This states that the logarithm of a product is equal to the sum of the logarithms.
• Quotient Rule: \( \log_{b}(\frac{x}{y}) = \log_{b}(x) - \log_{b}(y) \). This allows you to express the logarithm of a quotient as the difference of two logarithms.
• Power Rule: \( \log_{b}(x^r) = r \cdot \log_{b}(x) \). This is useful for expressions where the argument of the logarithm is raised to an exponent.

However, it's crucial to remember that these rules do **not** apply to sums or differences inside the logarithm. For instance, an expression like \( \log_{b}(x + y) \) cannot be directly simplified using these properties. It's already in its simplest form.

Algebraic Expressions

Algebraic expressions contain terms and operations involving numbers and variables. They can include addition, subtraction, multiplication, division, and even exponentiation.

When working with algebraic expressions within logarithms, such as \( \log_{6}(7m + 3q) \), it's important to recognize the type of expression you are dealing with. The expression inside the logarithm, \( 7m + 3q \), is a linear algebraic expression with two terms.

Unlike products and quotients, sums and differences within logarithmic expressions cannot be broken down using basic logarithmic properties. Logarithms of sums or differences, such as \( \log_{b}(x + y) \), require a different approach if any simplification is even possible. Sometimes, these expressions are already in their simplest form, and no further simplification can be made using the logarithm properties. This is why, in algebra, recognising the form and structure of expressions is critical.