Problem 65 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each expression without using a calculator. $$e^{\ln 125}$$

Step-by-Step Solution

Answer

The evaluated expression is 125

Step 1: Apply Logarithmic Property

We know from the properties of logarithms that \(e^{\ln a} = a\). Apply this property on the given expression \(e^{\ln 125}\), we get the value as 'a' which is 125 in this case.

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Step 6: Conclude with the answer

The evaluated expression is 125

Key Concepts

Exponential Functions

Exponential Functions are a crucial concept in mathematics, especially when dealing with growth and decay processes. Understanding these functions can help in solving a wide range of real-world problems. An exponential function is typically defined as any function that can be expressed in the form \[ f(x) = a \cdot b^x \]where 'a' is a constant, 'b' is the base of the exponent, and 'x' is the exponent itself. The most common base encountered in mathematics, particularly in calculus and natural processes, is the number 'e'.

• The number 'e' is an irrational constant approximately equal to 2.71828. It's the base of natural logarithms and has unique properties that make it ideal for continuous growth calculations.
• In exponential functions, when 'x' increases, the function value changes rapidly if 'b' is greater than 1, denoting growth. Conversely, if 'b' is between 0 and 1, it indicates decay.

Exponential functions are versatile and appear in various fields such as finance for calculating compound interest, in biology for population growth models, and in physics for radioactive decay. Their unique properties make them essential for understanding natural phenomena.

Natural Logarithms

Natural Logarithms are logarithms with the base 'e' and have vast applications in calculus and other branches of mathematics. The natural logarithm of a number 'x' is written as \[ \ln(x) \]and it answers the question "to what power must 'e' be raised, to get 'x'?". Here's why they are significant:

• The natural logarithm has several unique properties, including simplicity when dealing with rates of growth or decay. It turns multiplicative processes into additive ones, which are easier to handle.
• An important property of natural logarithms is the relationship \[ e^{\ln x} = x \]for positive 'x'. This property is a direct consequence of the definition of the logarithm base 'e'.

This property is particularly useful, as seen in the problem where \[ e^{\ln 125} = 125 \].Understanding natural logarithms allows us to decipher complex exponential equations into more manageable algebraic expressions.

Evaluating Expressions

Evaluating Expressions involves simplifying a mathematical phrase to find its value. This skill is fundamental in algebra and is repeated across different branches of mathematics. When evaluating expressions involving logarithms and exponents, it is important to know the properties that simplify the process.

• The key step in evaluating an expression like \[ e^{\ln 125} \]relies on identifying and using the property \[ e^\{\ln x\} = x \].Applying this property simplifies the expression immediately, providing the result without complicated calculations.
• Always remember that evaluating expressions accurately depends on understanding the relationships and properties inherent in the mathematical elements involved. For logarithms and exponential functions, properties that relate these functions simplify evaluations significantly.

To practice evaluating expressions, focus on recognizing patterns, knowing critical properties, and working through problems step-by-step. This will enhance proficiency and confidence in handling more complex mathematical operations.