Problem 280 - Precalculus

Question

For the following exercises, use properties of logarithms to evaluate without using a calculator. $$ 2 \log _{9}(3)-4 \log _{9}(3)+\log _{9}\left(\frac{1}{729}\right) $$

Step-by-Step Solution

Answer

The expression simplifies to -4.

Step 1: Apply Logarithm Properties to Combine Terms

First, recognize that the expression contains two logarithms with the same base, \( \log_{9}(3) \). This allows us to combine terms:\[ 2 \log_{9}(3) - 4 \log_{9}(3) = (2 - 4) \log_{9}(3) = -2 \log_{9}(3). \]

Step 2: Simplify the Logarithmic Expression

Next, replace the second part of the expression using the logarithm power property:\[ -2 \log_{9}(3) = \log_{9}(3^{-2}) = \log_{9}\left(\frac{1}{9}\right). \]

Step 3: Process Remaining Logarithms

Now, evaluate \( \log_{9}\left(\frac{1}{729}\right) \). Note that \( 729 = 9^3 \), so:\[ \log_{9}\left(\frac{1}{729}\right) = \log_{9}(9^{-3}) = -3 \log_{9}(9) = -3 \cdot 1 = -3. \]

Step 4: Combine Simplified Logarithms

Combine the results from Step 2 and Step 3:\[ \log_{9}\left(\frac{1}{9}\right) + \log_{9}\left(\frac{1}{729}\right) = (-1) + (-3) = -4. \]

Step 5: Final Result

The original expression simplifies to -4, considering all the simplified values and operations combined.

Key Concepts

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions is like cleaning up after a fun, messy art project. Just as you would organize your materials and tidy your workspace, you want to make your logarithm expression as neat and straightforward as possible. To achieve this, the first thing to do is to evaluate any similar terms. Let’s consider an expression with multiple terms having the same base within the logarithm, like \[ 2 \log_{9}(3) - 4 \log_{9}(3) + \log_{9}\left(\frac{1}{729}\right). \]By focusing on terms that share similar bases, you can often combine them or subtract one result from another, which simplifies the work considerably. Imagine if you had two cups of lemonade and you dumped one cup out. You’d be left with one less cup. In logarithms, when you have terms like \[ 2 \log_{9}(3) - 4 \log_{9}(3), \]you are ultimately determining what happens when two is subtracted from four, leaving you with \[ -2 \log_{9}(3). \]These steps in logical order guide you quickly towards balancing out your international classroom: the base remains the same, and you can focus on simplifying the expression further.

Logarithm Power Property

The power property of logarithms is all about giving a clear directive on how to use exponents within a logarithm effectively. This property states that \[ a \cdot \log_{b}(c) = \log_{b}(c^{a}). \]This rule can serve to either manipulate or reduce terms to make understanding complex logarithmic expressions easier.Imagine the power property as a scale where the external weight brings about balance. For instance, when you have \[ -2 \log_{9}(3), \]this becomes \[ \log_{9}(3^{-2}) = \log_{9}\left(\frac{1}{9}\right). \]Here, having the exponent on the outside transforms the logarithm into a more viable computation by housing it within the bigger picture (inside the logarithm).This property helps peel back the layers of the logarithm, unveiling simpler expressions more rapidly without unnecessary computations, so you can zoom through more exercises with ease.

Combining Logarithms

Combining logarithms is like cooking with your favorite family recipes; you bring together the best ingredients to create something enjoyable and efficient. The versatility of logarithms allows you to add or subtract individual parts, assuming they share the same base, into a single log expression, creating a more concise evaluation.For example, consider the expression\[ \log_{9}\left(\frac{1}{9}\right) + \log_{9}\left(\frac{1}{729}\right). \]Using properties of logarithms, these expressions combine into a single expression like so:

• Recognize the matching bases across separate logarithms.
• Combine them using the properties of logarithms.
• Evaluate the math simply with subtraction of exponents.

First, consider that \[ 9 = 9^1 \]and evaluating \[ \frac{1}{9} \]allows us to see that combining gives us \[ -1 \]Then, evaluating \[ \frac{1}{729} \]means you’d similarly compute \[ -3. \]Together, these become \[ -1 + (-3) = -4, \]simplifying your expression down magnificently to streamline the learning experience.