Problem 4 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{9}(9 x) $$

Step-by-Step Solution

Answer

The expanded form of \(\log _{9}(9x)\) is \(1 + \log _{9}(x)\).

Step 1: Isolate the Constants and Variables

We start by expanding the logarithm using the logarithmic property that states \(\log_b(a*c) = \log_b(a) + \log_b(c)\). In this case our \(a\) is 9 (a known base) and \(c\) is \(x\) which leaves us with \(\log _{9}(9) + \log _{9}(x)\).

Step 2: Evaluate Known Logarithms

Next, we can evaluate \(\log _{9}(9)\) which equals to 1. This is because any log base \(b\) of \(b\) equals 1. So the equation further simplifies to 1 + \(\log _{9}(x)\).

Step 3: Final Step

Put the result together to get the final, expanded form of the logarithmic expression. Which is \(1 + \log _{9}(x)\).

Key Concepts

Logarithmic Expression

Understanding logarithmic expressions is key in mathematics, especially when dealing with exponential relationships. A logarithmic expression, put simply, is the inverse of an exponential expression. It answers the question: to what exponent must the base be raised to yield a certain number? For instance, the logarithmic expression \(\log_b(x)\) means 'what power should we raise 'b' to, in order to get 'x'?'

When you see an expression like \(\log_{9}(9x)\), interpreting it might seem challenging at first. But remember, this simply means 'what power do we raise 9 to in order to get 9x?'. To answer this, there are properties of logarithms you can employ to make it easier to digest.

More specifically, logarithmic expressions can often be simplified or evaluated using rules such as the product rule—turning multiplication inside the log into addition outside of it—the quotient rule, and the power rule. These properties make it possible to break down complex logarithmic expressions into simple, solvable components.

Expand Logarithms

Expanding logarithms involves rewriting a single logarithmic expression into multiple simpler terms that are easier to evaluate. This process uses the logarithmic properties, such as the product, quotient, and power rules, to break down the expression.

For example, using the product rule, the expression \(\log_{b}(ac)\) can be expanded to \(\log_{b}(a) + \log_{b}(c)\), which is exactly what was done in the exercise with \(\log_{9}(9x)\). This rule essentially separates the multiplication inside the log into separate logs connected by addition. Utilizing these rules not only simplifies calculations but also aids in better understanding the nature of logarithmic relationships within equations.

• Product Rule: \(\log_{b}(ac) = \log_{b}(a) + \log_{b}(c)\)
• Quotient Rule: \(\log_{b}\left(\frac{a}{c}\right) = \log_{b}(a) - \log_{b}(c)\)
• Power Rule: \(\log_{b}(a^c) = c\cdot\log_{b}(a)\)

By committing these properties to memory, expanding logarithms becomes a straightforward, mechanical process.

Evaluate Logarithms

Evaluating logarithms is the process of simplifying a logarithmic expression to a specific numeral where possible, without necessarily using calculators. This requires a good grasp of basic logarithm rules and properties. For instance, the logarithm of a number, where the base is the same as the number itself, like \(\log_{9}(9)\), is always 1. This is one of the simplest evaluations you can encounter.

To evaluate logs without calculators, you'd typically look for exponents that fit the base—essentially finding exact values or simplifying the expression as much as possible. In the given exercise, \(\log_{9}(9)\) was evaluated as 1 because 9 raised to the power of 1 equals 9. After this, we can no longer simplify \(\log_{9}(x)\) without additional information about 'x'. Therefore, the log is left as is, resulting in the expression '1 + \(\log_{9}(x)\)'.

Always ensure to break down the expression using the aforementioned rules, and look for opportunities to simplify. Factors that are powers of the base, like 9 in this exercise, will often lead to clear evaluations. For logs without evident evaluations, rewrite them to clear up the expression and leave them in their simplest form.