Problem 42 - College Algebra

Question

For the following exercises, evaluate the base \(b\) logarithmic expression without using a calculator. $$\log _{3}\left(\frac{1}{27}\right)$$

Step-by-Step Solution

Answer

The logarithm \( \log_3\left(\frac{1}{27}\right) \) is \(-3\).

Step 1: Understand the Logarithm Definition

The expression \( \log_3\left(\frac{1}{27}\right) \) is asking to find the power to which the base 3 must be raised to get \( \frac{1}{27} \). This means finding an exponent \( x \) such that \( 3^x = \frac{1}{27} \).

Step 2: Rewrite the Fraction

Now, express \( \frac{1}{27} \) as a power of 3. Notice that \( 27 = 3^3 \) so \( \frac{1}{27} = 3^{-3} \).

Step 3: Set Up the Logarithmic Equation

Utilizing the rewritten expression, set the equation \( 3^x = 3^{-3} \). Our objective is to find \( x \).

Step 4: Solve for the Exponent

Since the bases are the same, equate the exponents: \( x = -3 \). Thus, the value of \( \log_3\left(\frac{1}{27}\right) \) is \(-3\).

Key Concepts

Understanding Exponents

When we talk about exponents, we're referring to the process of multiplying a number by itself a certain number of times. For instance, if you see a number like \(3^4\), it means you multiply 3 by itself 3 more times, resulting in 81. Exponents help simplify repeated multiplication, making complex calculations easier.

When bases are negative, exponents become a little more interesting. For example:

• Positive exponents: Like \(3^2 = 9\) indicate repeated multiplication.
• Negative exponents: Like \(3^{-2}= \frac{1}{9}\) indicate division or reciprocals and show how dividing the number can bring you to much smaller results.

Understanding exponents is crucial in logarithms since logarithms essentially reverse exponentiation, finding the exponent given a base and an outcome.

Base Conversion and its Importance

In mathematics, converting numbers to different bases can be quite handy! Base conversion is about expressing the same number using a different base. For example, the number 27 can be expressed in terms of base 3, as \(3^3\).

Base conversion is fundamental when solving logarithms because:

• It helps simplify logarithmic calculations by making expressions easier to handle.
• It allows for comparisons and simplification of mathematical equations, especially when the base and the number are related through common powers.

In our example, noticing that \(\frac{1}{27}\) could be expressed as \(3^{-3}\) was pivotal in solving the logarithm \(\log_3(\frac{1}{27})\), since it directly pointed us to the exponent needed for our base of 3.

Solving Equations with Logarithms

Logarithmic equations are solved by understanding and manipulating their exponential form. To solve the equation \(3^x = 3^{-3}\) we follow a few steps.

Firstly, with the logarithm equation in our problem:

• We express both parts with the same base, as both are based on 3 in this case.
• Next, we equate the exponents, because if \(a^n = a^m\), then \(n = m\), as long as \(a eq 0\).

This leads us to determine that \(x = -3\). Recognizing how to transition between the forms proves essential, simplifying steps significantly and making the problem accessible even without a calculator.

Logarithms naturally lend themselves to this kind of transformation because they define the relationship between a base and its power, helping solve the equation by focusing on what power of the base gives the provided result.