Problem 4 - Algebra and Trigonometry

Question

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 the given logarithmic expression is 1 + \(\log _{9}x\).

Step 1: Apply the product rule

We start by applying the product rule to the given logarithmic expression. So, we rewrite \(\log _{9}(9x)\) as \(\log _{9}9 + \log _{9}x\). This represents the sum of the logarithm of 9 and the logarithm of \(x\) base 9.

Step 2: Apply the power rule

We can now apply the power rule to simplify \(\log _{9}9\). The logarithm of any base to that same base equals 1. Therefore, \(\log _{9}9\) simplifies to 1.

Step 3: Write the final expression

By applying the power and product rules, the expression simplifies to 1 + \(\log _{9}x\). This is the expanded form of the original logarithmic expression, evaluated without using a calculator.

Key Concepts

Product Rule of Logarithms

The product rule of logarithms is a handy technique. It simplifies expressions that involve the logarithm of a product. To put it simply, the product rule states that the logarithm of a product is equal to the sum of the logarithms of its factors. This can be expressed in the formula: \[ \log_b(mn) = \log_b m + \log_b n \]In our example, we apply this rule to \( \log_9(9x) \). This turns into \( \log_9 9 + \log_9 x \). We've now split the expression into two simpler parts, making it easier to handle. Understanding this rule not only helps in solving problems but also gives insight into the behavior of logarithmic functions. Remember, being able to divide and conquer complicated expressions is a powerful math tool.

Power Rule of Logarithms

The power rule of logarithms is another critical tool for simplifying logarithmic expressions. This rule helps when you're dealing with exponents inside a logarithm. The power rule states that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base. So, in mathematical terms: \[ \log_b(m^n) = n \cdot \log_b m \]In the solution of \( \log_9(9x) \), the power rule applies to \( \log_9 9 \). Here, we're essentially looking at \( 9^1 \), which translates to 1 times the logarithm of 9 with base 9. This results in \( \log_9 9 = 1 \) since any number logarithmically compared to itself equals 1. It’s crucial to grasp this rule as it makes working with powers in logarithmic expressions much simpler.

Logarithmic Expression Expansion

Expanding logarithmic expressions involves using the properties of logarithms, like the product and power rules, to break down complex expressions into simpler parts. These rules help in transforming difficult-to-evaluate expressions into sums or differences that are more straightforward to handle. In the given exercise, after applying the product rule, we expanded \( \log_9(9x) \) into \( \log_9 9 + \log_9 x \).Next, by applying the power rule, we simplified \( \log_9 9 \) further to 1. This resulted in the expanded expression \( 1 + \log_9 x \). This form is much easier to interpret and use. Expansion is extremely useful when solving equations or integrating expressions in calculus. It reveals the structure of expressions and often leads to simpler forms. That's why mastering how to expand logarithmic expressions is indispensable in advanced mathematics.