Problem 163 - Precalculus

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 simplified expression is 14.125.

Step 1: Understand the Problem

The expression given is \(e^{\ln (10.125)} + 4\). We need to simplify this expression using the properties of logarithms and exponentials.

Step 2: Apply Logarithmic and Exponential Properties

Recall that the logarithmic identity states \(e^{\ln(x)} = x\) for any positive \(x\). This is because the exponential function and the natural logarithm are inverse operations.

Step 3: Simplify Using the Property

Apply the identity from Step 2 to the given expression: \(e^{\ln (10.125)} = 10.125\). The expression simplifies to \(10.125 + 4\).

Step 4: Perform the Arithmetic

Calculate \(10.125 + 4\). The result is \(14.125\).

Key Concepts

Exponential Functions

Exponential functions are mathematical functions of the form \(f(x) = a^x\), where 'a' is a positive constant and 'x' is the variable exponent. They are known for their rapid growth rate, as the function's output increases exponentially as 'x' increases. A special type of exponential function is \(e^x\), where 'e' is Euler's number (approximately 2.71828). This function is commonly used in natural logarithms and calculus.

Exponential functions are found in various real-world scenarios:

• Population growth models
• Compound interest calculations
• Radioactive decay

Understanding and working with exponential functions is essential for solving problems related to exponential growth and decay.

Logarithmic Identities

Logarithmic identities are equations that provide relationships between logarithms of different quantities. A vital identity to know is for natural logarithms and the exponential function: \(e^{\ln(x)} = x\). This identity is based on the inverse relationship between an exponential function and its corresponding logarithm, the natural logarithm \(\ln(x)\).

Key logarithmic identities include:

• \(\ln(ab) = \ln(a) + \ln(b)\)
• \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)
• \(\ln(a^b) = b \cdot \ln(a)\)

Logarithmic identities simplify complex expressions and are essential for solving equations where variables are in the exponents. They make calculations more manageable, especially when dealing with multiplicative relationships.

Simplification of Expressions

The simplification of expressions involves reducing them to a simpler or more easily understandable form. This is often achieved through various algebraic techniques, using known identities or mathematical properties. In the context of the original exercise, the use of the identity \(e^{\ln(x)} = x\) simplifies more complex expressions.

For simplification:

• Identify and apply applicable mathematical identities or known properties.
• Carry out arithmetic operations step-by-step to ensure accuracy.

In the original problem, we had \(e^{\ln(10.125)} + 4\). By the identity, \(e^{\ln(10.125)}\) simplifies directly to 10.125. Adding the 4 gives the final simplified result of 14.125. This process saves time and reduces the potential for errors in complex calculations, making it a valuable skill in mathematical problem-solving.