Problem 41 - College Algebra

Question

For the following exercises, use the definition of common and natural logarithms to simplify. $$ e^{\ln (10.125)}+4 $$

Step-by-Step Solution

Answer

The expression simplifies to 14.125.

Step 1: Understand the Problem

We need to simplify the expression \( e^{\ln (10.125)} + 4 \). We will use properties of logarithms and exponents to achieve this.

Step 2: Simplify the Exponential

Recall the property of logarithms and exponents: \( e^{\ln(x)} = x \). Therefore, \( e^{\ln(10.125)} = 10.125 \).

Step 3: Substitute Simplified Value

Replace \( e^{\ln(10.125)} \) with \( 10.125 \): the expression becomes \( 10.125 + 4 \).

Step 4: Perform the Arithmetic

Add the numbers together: \( 10.125 + 4 = 14.125 \). Thus, the simplified expression is 14.125.

Key Concepts

Common Logarithms

Logarithms are mathematical operations that help us find out how many times we need to multiply a number (called the base) by itself to get another number. Common logarithms are logarithms that use the base 10. They are often written as \( \log(x) \), but sometimes the base 10 is implied, without writing it.

• For example, \( \log_{10}(100) = 2 \) because 10 multiplied by itself twice is 100.
• Common logarithms are helpful in a variety of real-world situations, like calculating interest rates and measuring the intensity of earthquakes or sound.

Understanding common logarithms is crucial when dealing with exponential equations and simplifying expressions like \( e^{\ln(x)} \). These can often be broken down using properties of logarithms, resulting in simpler numbers to work with.

Properties of Exponents

Exponents represent numbers repeated in multiplication. For example, \( x^a \) means we multiply \( x \) by itself \( a \) times. Recognizing and using the properties of exponents can make complex expressions much easier to simplify. Let's look at a few important properties:

• Power of a Power: \( (x^a)^b = x^{ab} \)
• Product of Powers: \( x^a \cdot x^b = x^{a+b} \)
• Quotient of Powers: \( \frac{x^a}{x^b} = x^{a-b} \)

By understanding these properties, you can transform complex exponential equations into simpler, more manageable forms. An example used in the given exercise is \( e^{\ln(x)} = x \), which utilizes the understanding of exponential and logarithmic relationships.

Properties of Logarithms

Logarithms have several inherent properties that make them incredibly versatile and useful for solving equations and simplifying terms. Grasping these properties allows us to manipulate logarithmic expressions intelligently. Here are a few key properties:

• Product Rule: \( \log_b(xy) = \log_b(x) + \log_b(y) \)
• Quotient Rule: \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \)
• Power Rule: \( \log_b(x^y) = y \cdot \log_b(x) \)

In the exercise, the property \( e^{\ln(x)} = x \) was used to simplify the expression quickly. Understanding the natural pairing of logarithms with exponents, particularly natural logs (with base \( e \)), can significantly reduce the complexity of an equation and bring clarity to solutions.