Problem 32 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Use the laws of logarithms to solve the equation. $$\log _{3} 27=2 x$$

Step-by-Step Solution

Answer

The given equation is \(\log_{3} 27 = 2x\). Rewrite it as an exponent equation: \(3^{2x} = 27\). Simplify by writing 27 as a power of 3: \(3^{2x} = 3^3\). Equate the exponents: \(2x = 3\). Solve for x: \(x = \frac{3}{2}\).

Step 1: Understand the equation

We need to solve for x in the equation \(\log_{3} 27 = 2x\).

Step 2: Use the exponent rule of logarithms

Rewrite the equation as an exponent equation using the rule \(\log_{b} a = x\) is equivalent to \(b^x = a\). So, the equation becomes: \(3^{2x} = 27\).

Step 3: Simplify the equation

We can rewrite 27 in terms of a power of 3. Since \(3^3 = 27\), the equation becomes: \(3^{2x} = 3^3\).

Step 4: Equate the exponents

Now that our equation is in the form of \(b^x = b^y\), we can equate the exponents: \(2x = 3\).

Step 5: Solve for x

Divide both sides of the equation by 2: \(x = \frac{3}{2}\). The solution to the equation \(\log_{3} 27 = 2x\) is \(x = \frac{3}{2}\).

Key Concepts

Exponentiation

Exponentiation is a fundamental mathematical operation where a number, the base, is raised to the power of another number, the exponent. In simpler terms, it's like multiplying a number by itself a certain number of times. In our equation, we are dealing with base 3 raised to an exponent.Look at the base-exponent pair in the original equation \(3^{2x} = 27\). The goal is to express both sides with the same base so we can equate the exponents. Here, the number 27 can be written as a power of 3, which is \(3^3\).

• The base in both sides is 3, an important simplification.
• This enables equating the exponents directly, simplifying further solving steps.

Exponentiation can be powerful for simplifying equations, especially when numbers can be expressed in similar terms, reducing complex calculations to more manageable comparisons of exponents.

Equation Solving

Equation solving is the process of finding the unknown variable that satisfies the given equation. Here, the task is to find the value of \(x\) such that the equation holds true.In our particular exercise, the sequence of steps efficiently hones in on solving the unknown:

• Convert logarithmic expression to an equivalent exponential form, producing \(3^{2x} = 27\).
• Rewrite both sides with a common base for simpler comparative solving \(3^{2x} = 3^3\).
• Equate the exponents, direct result from having the same base: \(2x = 3\).
• Solve the simple algebraic equation for \(x\), dividing both sides by 2: \(x = \frac{3}{2}\).

The problem-solving method adopted efficiently narrows down the steps by utilizing properties of equality for exponents, emphasizing the importance of clarity and organization in finding solutions.

Properties of Logarithms

Understanding the properties of logarithms makes dealing with logarithmic equations straightforward and logical. Logarithms are essentially the inverse operation of exponentiation, designed to find exponents based on bases.Consider the basic property used in this problem: \(\log_{b} a = x\) implies \(b^x = a\). This conversion between logarithmic and exponential form is a vital technique used to simplify the given equation into something more solvable.

• It turns complex log equations into manageable exponential ones.
• It's a key principle in changing the perspective of a problem, making it solvable through exponentiation.

Using these properties of logarithms is crucial in problems of this type because they allow us to work with numbers in a way that aligns with their intuitive arithmetic properties, such as multiplication turning into addition, and exponentiation easing comparative solving.