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Chapter 12: Problem 80

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

Short Answer

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

Step by step solution

01

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, \(ln(e) = 1\).
02

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 \(\log(1)\).
03

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 \(\log(1) = 0\) .
04

Substituting \(\log(1)\) with 0

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

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Most popular questions from this chapter

The percentage of adult height attained by a girl who is \(x\) years old can be modeled by $$f(x)=62+35 \log (x-4)$$ where \(x\) represents the girl's age (from 5 to 15 ) and \(f(x)\) represents the percentage of her adult height. Use the formula to solve Exercises. Round answers to the nearest tenth of a percent. Approximately what percentage of her adult height has a girl attained at age ten?

Without using a calculator, determine which is the greater number: \(\log _{4} 60\) or \(\log _{3} 40\)

$$\text { Solve: } x-2(3 x-2)>2 x-3$$

Graph: \(5 x-2 y>10\)

Without using a calculator, find the exact value of \(\log _{4}\left[\log _{3}\left(\log _{2} 8\right)\right]\)
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