Problem 80 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each expression without using a calculator. $$\log (\ln e)$$

Step-by-Step Solution

Answer

The short answer is, \(\log(\ln(e)) = 0\) .

Step 1: Understanding the concept of natural log

The natural logarithm, often denoted as \(ln\), is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.718281. We use natural logarithms when we are dealing with anything that grows in a 'natural' way. The function \(ln(e)\) will give us the power to which we must raise \(e\) to achieve \(e\). Since any number to the power of 1 is the number itself, \(ln(e) = 1\).

Step 2: Substituting \(ln(e)\) with 1

In the given expression, we replace \(ln(e)\) with 1 (from Step 1). The expression \(ln(e)\) becomes 1, so our expression becomes \(\log(1)\).

Step 3: Understanding the concept of base 10 logarithm

The logarithm without an indicated base is called the common logarithm and has a base of 10. The logarithm \(\log(1)\) to the base 10 refers to the exponent that the base (10) must be raised to, to get the argument (1). Since any number to the power of 0 is 1, we have \(\log(1) = 0\) .

Step 4: Substituting \(\log(1)\) with 0

In the expression, replace \(\log(1)\) with 0 (from Step 3). So our final answer is 0.