Problem 16 - Precalculus

Question

Evaluate the expression. \(\log _{3}(27)\)

Step-by-Step Solution

Answer

The value of \(\log_{3}(27)\) is 3.

Step 1: Understand the Logarithm

The expression \(\log_{3}(27)\) is asking for the exponent to which the base 3 must be raised to obtain 27. This can be written as \(3^x = 27\).

Step 2: Write 27 as a Power of 3

Notice that 27 is a power of 3. In exponential terms, 27 can be written as \(3^3\). So the equation becomes \(3^x = 3^3\).

Step 3: Equate the Exponents

Since the bases on both sides of the equation are the same (3), the exponents must be equal. Therefore, solve for \(x\) to find that \(x = 3\).

Step 4: Conclusion

Conclude that \(\log_{3}(27) = 3\), because raising 3 to the power of 3 gives 27.

Key Concepts

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In its general form, it is expressed as \( f(x) = a^x \), where \( a \) is a positive constant, and \( x \) can be any real number. Exponential functions are fundamental in mathematics due to their unique properties and applications in various fields like finance, biology, and physics.

• Growth and Decay: Depending on the context, these functions can model growth (like population increase) or decay (radioactive decay).
• Continuous Growth Rate: They describe processes that grow at a constant percentage rate over time.

In the original exercise, recognizing that 27 can be expressed as \(3^3\) is an application of exponential functions. It highlights how exponential functions can simplify complex equations by expressing numbers as powers of a base.

Base of a Logarithm

The base of a logarithm refers to the constant number that is raised to a certain power to produce another number. In the expression \( \log_{b}(x) \), \( b \) is the base of the logarithm. This base must always be a positive number other than 1. Common bases are 10 for common logarithms (log) and \( e \) for natural logarithms (ln).

• Understanding the Base: The base determines the scaling of the logarithmic function and is crucial in solving logarithmic equations.
• Relationship to Exponential Functions: The logarithm is the inverse operation of the exponential function; it "undoes" the exponentiation.

In our exercise, the base of the logarithm is 3, indicating we are looking for the power that 3 needs to be raised to make 27. Recognizing the base is key to transitioning between exponential and logarithmic forms.

Logarithmic Equation

A logarithmic equation involves the logarithm of an unknown quantity. Solving these equations often requires converting them into exponential form. For example, the equation \( \log_{b}(y) = x \) can be rewritten as \( b^x = y \). This transformation helps in simplifying the problem:

• Solving Technique: By converting a logarithmic equation to its equivalent exponential form, we can often find the solution more easily.
• Using Known Values: If the solution involves known powers of the base, identifying this can streamline the process.

The problem \( \log_{3}(27) = x \) was solved by recognizing it as \( 3^x = 27 \), thus simplifying the solution by equating the exponents.

Properties of Exponents

Understanding the properties of exponents is crucial for solving equations involving powers. These properties allow us to manipulate and simplify expressions efficiently:

• Product of Powers: \( a^m \cdot a^n = a^{m+n} \)
• Power of a Power: \( (a^m)^n = a^{m \cdot n} \)
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \) where \( a eq 0 \)

In the exercise, once 27 was rewritten as \( 3^3 \), we equated the exponents since both sides shared the same base. This direct application of the power of a power property \( (a^m)^n = a^{m \cdot n} \) simplifies many math problems, allowing for efficient problem-solving and a better grasp of exponential relationships.