Problem 49 - Algebra 2 and Trigonometry

Question

In \(27-56,\) evaluate each logarithmic expression. Show all work. $$ \log _{\frac{1}{3}} 27 $$

Step-by-Step Solution

Answer

\( \log_{\frac{1}{3}} 27 = -3 \)

Step 1: Understanding the Logarithm Definition

The logarithm \( \log_b a \) asks the question: "To what power must the base \( b \) be raised, to produce the number \( a \)?" In this case, we need to determine \( x \) such that \( \left(\frac{1}{3}\right)^x = 27 \).

Step 2: Express the Equation with Powers

Rewrite 27 as a power of 3 to align the bases. We know that \( 27 = 3^3 \). Therefore, the equation becomes \( \left(\frac{1}{3}\right)^x = 3^3 \).

Step 3: Transform the Base Fraction

Rewrite \( \left(\frac{1}{3}\right)^x \) using negative exponents: \( \left(\frac{1}{3}\right) = 3^{-1} \). This changes the equation to \( (3^{-1})^x = 3^3 \).

Step 4: Simplify Using Power Rules

Apply the power of a power rule \((a^m)^n = a^{m imes n}\). Using this on \( (3^{-1})^x \), we get \( 3^{-x} = 3^3 \).

Step 5: Equate the Exponents

Since the bases are the same, we equate the exponents: \(-x = 3\).

Step 6: Solve for \( x \)

Solve the simple linear equation \(-x = 3\) for \( x \). Multiplying both sides by \(-1\), we find \( x = -3 \).

Key Concepts

Powers of Numbers

Powers of numbers, also known as indices, is essentially repeated multiplication of a number by itself. When we say a number is raised to a power, it is written as \( a^b \), where \( a \) is the base and \( b \) is the exponent or power. For example, \( 3^3 \) means that 3 is multiplied by itself three times: \( 3 \times 3 \times 3 = 27 \).
Knowing how to express numbers as powers is particularly helpful when dealing with logarithms, as it allows you to easily match bases and simplify problems. Understanding this concept simplifies many mathematical operations, making tasks like comparing sizes or solving equations more manageable.
Powers of numbers streamline calculations and are foundational in many areas of mathematics, helping to convey large numbers or small fractions succinctly.

Exponents

Exponents indicate how many times a base number is used as a factor in a multiplication operation. Simply put, an exponent is the tiny number hovering to the top right of the base number.
For instance, in the expression \( 2^4 \), 2 is the base and 4 is the exponent, telling us to multiply two by itself four times: \( 2 \times 2 \times 2 \times 2 = 16 \). This handy shorthand is extremely useful for simplifying repetitive multiplications.
Furthermore, understanding exponents is crucial for working with different mathematical concepts like roots, logarithms, and polynomial equations. Mastering exponents allows you to express and solve various forms of equations effortlessly.

Negative Exponents

Negative exponents demonstrate an interesting rule in mathematics. They essentially represent the reciprocal of the base raised to the absolute value of the exponent.
In practice, \( a^{-b} \) is equivalent to \( \frac{1}{a^b} \). For example, \( 3^{-1} \) equals \( \frac{1}{3} \), and further expanding that concept, \( (\frac{1}{3})^2 = \frac{1}{9} \). This property helps convert division into multiplication, making equations often easier to handle.
In the context of logarithms, using negative exponents allows for switching between bases and simplifying expressions. This is crucial in understanding how to write expressions in a form that is easier to compare and solve—such as moving between \( (\frac{1}{3})^x \) to \( 3^{-x} \).
Mastering negative exponents reveals a new dimension of flexibility in manipulating and solving algebraic expressions.

Solving Equations

The art of solving equations involves finding the value of the variable that makes the equation true. For logarithmic equations, the process often involves converting the logarithm into an exponential form and equating the powers.
Consider the equation \( \log_b a = x \), which can be rewritten as \( b^x = a \). Utilizing this transformation allows you to solve equations by aligning bases and comparing exponents.
For instance, when faced with \( (\frac{1}{3})^x = 27 \), recognizing that \( 27 = 3^3 \) simplifies the equation considerably after converting both sides to a power with the base of 3. This transformation benefits from aligning powers and equates the exponent values.
Breaking down equations and employing methods such as equating the exponents when the bases are the same is an effective way to simplify and solve complex equations.