Problem 66 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each expression without using a calculator. $$e^{\ln 300}$$

Step-by-Step Solution

Answer

The evaluation of the expression \(e^{\ln 300}\) is 300.

Step 1: Identifying the expression

The given expression is \(e^{\ln 300}\). In this expression, the base 'e' of the exponential function is the same as the base of the natural logarithm, indicating these are inverse functions.

Step 2: Applying the law of exponentials

According to the laws of exponentials, the exponential and natural logarithms being inverse functions of each other, the expression \(e^{\ln 300}\) simplifies directly to just 300. This could also be seen as an application of the identity \(a^{\log_a x} = x\), where 'a' is the common base of 'e'.

Key Concepts

Natural Logarithms

Natural logarithms are a special type of logarithm with a base of the mathematical constant 'e', which is approximately 2.718. The notation for natural logarithms is \( \ln \), and it solves for which power we must raise 'e' to get a certain number. For instance, if you have \( \ln x = y \), this means that \( e^y = x \). Natural logarithms are widely used in calculus and complex mathematical models, because the base 'e' is often more convenient than other bases when working with continuous growth or compound interest formulas.
Natural logarithms are the inverse of exponential functions that have the same base. This means they effectively "undo" the exponential function. For example, if you have \( e^{\ln x} \), it will simplify directly to \( x \), because the exponential function and the natural logarithm "cancel out" each other's effects. This property is very useful in simplifying complex mathematical expressions.

Inverse Functions

Inverse functions are paired functions that, when composed, undo each other's effects. When you have a function \( f(x) \) and its inverse \( f^{-1}(x) \), applying one followed by the other returns you to your original input: \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). Exponential functions and logarithms provide classic examples of inverses in mathematics.
The natural logarithm and its corresponding exponential function \( e^x \) are inverse functions. That is why, in problems like \( e^{\ln 300} \), the two operations cancel each other out, leaving just \( 300 \). This happens because the natural logarithm \( \ln \) essentially asks "What power should I raise 'e' to, to get my input?" Applying the exponential function \( e \) right afterward performs exactly that exponentiation, reversing the process right back to the original input number.

Laws of Exponents

The laws of exponents are important mathematical rules that simplify expressions involving powers. They make calculations consistent and reliable across different contexts. One of the key laws is \( a^{\log_a x} = x \), which is integral in solving exponential and logarithmic expressions.
This particular law applies anytime you have a base 'a' raised to the power of a logarithm with the same base. In our expression \( e^{\ln 300} \), 'e' is the base, and the natural logarithm uses the same 'e' as its base. This simplifies to \( 300 \) directly, showcasing the power and elegance of these laws.
Other fundamental rules include the product rule \( a^m \cdot a^n = a^{m+n} \), the quotient rule \( \frac{a^m}{a^n} = a^{m-n} \), and the power rule \( (a^m)^n = a^{m\cdot n} \). These rules allow for the easy manipulation and simplification of exponential expressions, aiding calculations in scientific and engineering disciplines.