Problem 278 - Precalculus

Question

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

Step-by-Step Solution

Answer

The final answer is -5.

Step 1: Use the Quotient Property of Logarithms

The problem given is \( \log _{3}\left(\frac{1}{9}\right) - 3 \log _{3}(3) \). First, we'll simplify \( \log _{3}\left(\frac{1}{9}\right) \). The quotient property states \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \), allowing us to rewrite \( \log _{3}\left(\frac{1}{9}\right) \) as \( \log_3(1) - \log_3(9) \). Since \( \log_3(1) = 0 \) because any number to the power of 0 is 1, the expression becomes \( -\log_3(9) \).

Step 2: Simplify \( \log_3(9) \)

Recognize that \( 9 = 3^2 \). So, \( \log_3(9) = \log_3(3^2) \). Using the power property of logarithms \( \log_b(M^n) = n\log_b(M) \), we have \( \log_3(3^2) = 2\log_3(3) \). Therefore, \( -\log_3(9) = -2\log_3(3) \).

Step 3: Simplify \( 3 \log_3(3) \)

Understand that \( \log_3(3) = 1 \) because 3 is the base raised to the power of 1 to get 3. Therefore, \( 3\log_3(3) = 3 \times 1 = 3 \).

Step 4: Combine and Simplify the Expression

Substitute \(-2\log_3(3)\) and \(3\log_3(3)\) into the original expression: \(-2\log_3(3) - 3\log_3(3) \). This simplifies to \(-2 - 3 = -5 \).

Key Concepts

Properties of Logarithms

Logarithms can appear tricky at first, but understanding their properties makes them much easier to work with. The properties of logarithms give us handy tools to simplify and evaluate expressions. These properties include:

• The product property: If you multiply two numbers inside a logarithm, it becomes the sum of two separate logarithms. $$ \log_b(M \cdot N) = \log_b(M) + \log_b(N) $$
• The quotient property: When you divide inside a logarithm, it converts to subtraction. $$ \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) $$
• The power property: If you raise a number inside a logarithm to a power, you can bring the exponent in front as a multiplier. $$ \log_b(M^n) = n \cdot \log_b(M) $$

Each property allows us to break down complex logarithmic expressions into simpler parts, just like we did in the given exercise.

Quotient Property

The quotient property of logarithms is particularly useful when dealing with fractions inside a log. This property states that the logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator. This is written as:\[\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\]In our original step-by-step solution, we used this property to break down the expression \[\log_3\left(\frac{1}{9}\right)\]We expressed it as\[\log_3(1) - \log_3(9)\]Because we know that \(\log_3(1) = 0\) (since any base raised to 0 is 1), this simplifies neatly to \(-\log_3(9)\). This property is a vital tool when simplifying logarithmic expressions.

Power Property

The power property of logarithms is another fantastic tool that helps simplify expressions where numbers are raised to a power inside the logarithm. It’s particularly handy when you know the base and the exponent form a power. It states:\[\log_b(M^n) = n \cdot \log_b(M)\]Applying this in our exercise allowed us to take \(\log_3(9)\) and rewrite it. Recognizing that 9 is actually \(3^2\), we applied the power property to get:\[\log_3(3^2) = 2 \cdot \log_3(3)\]This gives us \(2 \times 1 = 2\), because \(\log_3(3)\) is 1. Hence, \(-\log_3(9)\) could be simplified step-by-step to \(-2 \cdot \log_3(3)\), again leading us closer to solving the expression.

Evaluating Logarithms Without a Calculator

Having a good grasp of logarithm properties enables you to solve logarithmic expressions even without the use of a calculator. Evaluating logarithms without technology boils down to understanding and applying properties like the power and quotient properties to simplify the expressions.To solve a logarithmic expression by hand:

• Look for opportunities to rewrite numbers using familiar bases, like expressing 9 as \(3^2\).
• Utilize the power property to turn powers into products.
• Use the quotient property to break fractions into easier parts.
• Combine simplified parts by adding or subtracting, depending on the operation given.

For instance, in our example expression \(\log_3\left(\frac{1}{9}\right) - 3 \log_3(3)\), breaking down each component allowed us to pinpoint that the expression simplified to \(-5\). Now, whenever you approach log problems, remember to think about these operations, helping you solve without a calculator.