Problem 33 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Use the laws of logarithms to solve the equation. $$\log _{x} 10^{3}=3$$

Step-by-Step Solution

Answer

Using the properties of logarithms, we can rewrite the given equation \(\log _{x} 10^{3}=3\) in exponential form as \(x^3 = 10^3\). To solve for x, we can take the cube root of both sides of the equation and simplify, giving us the solution \(x = 10\).

Step 1: Convert to exponential form

We have \(\log _{x} 10^{3}=3\). Using the properties of logarithms, we can rewrite this equation as an exponential form: \(x^3 = 10^3\). Step 2: Solve for x

Step 2: Find the value of x

To find the value of x, we now need to solve the equation \(x^3=10^3\). We can get the cube root of both sides of the equation to isolate x: \(x =\sqrt[3]{10^3}\). Step 3: Simplify and calculate the value of x

Step 3: Simplify and calculate

Simplifying the expression, we get \(x = 10\). The value of x that makes the given equation true is \(x = 10\).

Key Concepts

Laws of Logarithms

Understanding the laws of logarithms is key to solving logarithmic equations. These laws are mathematical rules that allow us to work with logarithms more easily. One of the fundamental properties is that the logarithm of a power, such as \(\log_b(a^n)\), can be expressed as the exponent times the logarithm of the base, which in this case would be \(n \cdot \log_b(a)\).

Another important law states that the logarithm of a product can be written as the sum of the logarithms of the individual factors, meaning that \(\log_b(mn) = \log_b(m) + \log_b(n)\). Conversely, the logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator, expressed as \(\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)\). These rules are essential when breaking down complex logarithmic expressions into more manageable pieces that can be solved.

Exponential Form

When solving logarithmic equations, converting the log form to exponential form often provides a clearer path to the solution. This process utilizes the definition of logarithms: if \(\log_b(a) = c\), then the equivalent exponential form is \(b^c = a\).

To illustrate, in the given example equation \(\log_x(10^3) = 3\), we use the definition to convert it to \(x^3 = 10^3\), now represented in exponential form. This conversion is a turning point in solving the original problem as it sets the groundwork for using arithmetic operations, like taking the cube root, to find the value of x.

Cube Root

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. In mathematical notation, the cube root of a number a is denoted as \(\sqrt[3]{a}\). Cube roots are a special case of nth roots, where in this case n is 3.

In the context of our logarithmic equation, once we have \(x^3 = 10^3\), we need to isolate x. Extracting the cube root of both sides, \(\sqrt[3]{x^3} = \sqrt[3]{10^3}\), simplifies to \(x = 10\), as the cube root and the cube power are inverse operations. Understanding the concept of roots, particularly cube roots, is indispensable in order to reverse the process of cubing and to solve equations involving third powers.