1.3 Natural Logarithms

Your investment is earning compound interest of 6% per year. You wonder how long it will take to double in value. Answering this question requires you to solve an equation in which the unknown variable is located in the exponent instead of the base position. To find the answer, you need logarithms, which are reviewed in this section and will be applied in Chapter 9.

Logarithms

A logarithm is defined as the exponent to which a base must be raised to produce a particular power. There are two values for the base that are most often used: 10 and [latex]e[/latex].

A Base Value of 10. This is referred to as a common logarithm.

For example, since we have [latex]10^2 = 100[/latex], we have that the common logarithm of 100 is 2. This is written as [latex]\log_{10}(100) = 2[/latex], or just [latex]\log (100) = 2[/latex] (you can assume the base of 10 if the base is not written).

The way these expressions are read is as follows:

[latex]\log_{10}(100) = 2[/latex]: “log base ten of 100 is 2”

[latex]\log (100) = 2[/latex]: “log of 100 is 2”

The format for a common logarithm can then be summarized as [latex]\log (\text{power of }10) = \text{exponent of }10[/latex].

For example, if you have [latex]10^x = y[/latex], then [latex]\log (y) = x[/latex].

A Base Value of [latex]e[/latex]. This is referred to as a natural logarithm.

In mathematics, there is a known constant [latex]e[/latex], which has nonterminating decimals and has an approximate value of [latex]e = 2.71828182845[/latex].

For example, since [latex]e^3 = 20.085537[/latex], we have that [latex]3[/latex] is the natural logarithm of [latex]20.085537[/latex] and you write this as [latex]\log_e (20.085537) = 3[/latex] or [latex]\ln (20.085537) = 3[/latex].

The way these expressions are read is as follows:

[latex]\log_e (20.085537) = 3[/latex]: “log base [latex]e[/latex] of 20.085537 is 3″

[latex]\ln (20.085537) = 3[/latex]: “ln (pronounced as “lawn”) of 20.085537 is 3″

The format for a natural logarithm is then summarized as: [latex]\ln (\text{power of }e) = \text{exponent of }e[/latex].

For example, since we have [latex]e^4 = 54.59815[/latex], then [latex]\ln (54.59815) = 4[/latex].

Common logarithms have been used in the past when calculators were not equipped with power functions. However, with the advent of computers and advanced calculators that have power functions, the natural logarithm is now the most commonly used logarithm. From here on, this textbook focuses on natural logarithms only.

Properties of Natural Logarithms

Natural logarithms possess six properties:

  1. The natural logarithm of 1 is zero.
    This is because, if 1 is the power and 0 is the exponent, then you have [latex]e^0 = 1[/latex].
  2. The natural logarithm of any number greater than 1 is a positive number.
    For example, the natural logarithm of 2 is 0.693147 because [latex]e^{0.693147} = 2[/latex].
  3. The natural logarithm of any number less than 1 is a negative number.
    For example, the natural logarithm of 0.5 is −0.693147 because [latex]e^{−0.693147} = 0.5[/latex].
  4. A natural logarithm cannot be applied to a number that is less than or equal to zero.
    Since [latex]e[/latex] is a positive number with an exponent, there is no value of the exponent that can produce a power of zero. As well, it is impossible to produce a negative number when the base is positive.
  5. The natural logarithm of the quotient of two positive numbers is [latex]\ln \left( {\frac{x}{y}} \right) = \ln \left( x \right) - \ln \left( y \right)[/latex].
    For example,

[latex]\ln \left( {\frac{20000}{15000}} \right) = \ln \left( 20000 \right) - \ln \left( 15000 \right)[/latex]

  1. The natural logarithm of a power of a positive base is [latex]\ln x^y =y\ln x[/latex].
    This property allows you to bring the exponent down into the base.
    For example, to solve [latex]1.05^x=0.292741[/latex] for [latex]x[/latex], we can apply [latex]\ln[/latex] to bring the unknown variable down from the exponent:

[latex]\begin{align*} 1.05^x=0.292741&\Rightarrow \ln 1.05^x=\ln 0.292741\\ &\Rightarrow x\ln 1.05=\ln 0.292741\\ &\Rightarrow x=\frac{\ln 0.292741}{\ln 1.05}\approx -25.178579 \end{align*}[/latex]

Paths To Success

You do not have to memorize the mathematical constant value of [latex]e[/latex]. If you need to recall this value, use an exponent of 1 and access the [latex]e^x[/latex] function on your calculator. Hence, [latex]e^1 = 2.71828182845[/latex].

Concept Check

Try out your understanding of properties of logarithms:

MathMatize: Properties of logarithms

Give It Some Thought

For each of the following powers, determine if the natural logarithm is positive, negative, zero, or impossible.

a. 2.3       b. 1        c. 0.45       d. 0.97         e. −2         f. 4.83       g. 0

Give it Some Thought Solutions:

a. positive (property 2); b. zero (Property 1); c. negative (property 3)

d. negative (property 3); e. impossible (Property 4); f. positive (property 2); g. impossible (Property 4)

Example 1.3 A: Applying Natural Logarithms and Properties

Solve the first two questions using your calculator. For the next two questions, rewrite using the applicable property.

a. [latex]\ln(2.035)[/latex] b. [latex]\ln(0.3987)[/latex] c. [latex]\ln\left(\frac{10,000}{6,250} \right)[/latex] d. [latex]\ln(1.035)^{12}[/latex]

Answer:

a. [latex]\ln(2.035)=0.710496[/latex]

b. [latex]\ln(0.3987)=−0.919546[/latex]

c. [latex]\ln\left(\frac{10,000}{6,250} \right)=\ln (10,000)-\ln (6,250)[/latex]

d. [latex]\ln(1.035)^{12}=12\ln(1.035)[/latex]

 

Exercises

Mechanics

Using your scientific calculator, evaluate each of the following. Write your answer using the exponential equation format [latex]e^x =[/latex] power. Round your answers to six decimals.

2. [latex]\ln(0.7445)[/latex]

3. [latex]\ln(1.83921)[/latex]

4. [latex]\ln(13.2746)[/latex]

5. [latex]\ln(0.128555)[/latex]

Applications

In questions 9. – 11., rewrite each of the following using the fifth property of natural logarithms, where [latex]\ln \left( x^y \right) = y(\ln x)[/latex].

6. [latex]\ln\left(\frac{28500}{19250}\right)[/latex]

7. [latex]\ln\left(\frac{100000}{10000}\right)[/latex]

8. [latex]In questions 9. – 11., rewrite each of the following using the sixth property of natural logarithms, where [latex]\ln\left(x^y\right)=y\ln x[/latex].

9. [latex]\ln[(1.02)^23][/latex]

10. [latex]\ln[(1.01275)^41][/latex]

11. [latex]\ln[(1.046)^34][/latex]

In questions 12. – 14., evaluate [latex]\frac{\ln\left( \frac{FV}{PV} \right)}{\ln\left( 1+i \right)}[/latex] by substituting the given values.

[latex]FV[/latex] [latex]PV[/latex] [latex]i[/latex]
12. $78,230 $25,422 0.0225
13. $233,120 $91,450 0.0425
14. $18,974 $8,495 0.02175

Challenge, Critical Thinking, & Other Applications

In questions 15. – 16., rearrange [latex]FV = PV(1 +i)^N[/latex] to solve for [latex]N[/latex], then determine the value of [latex]N[/latex] by substituting in the given values.

[latex]FV[/latex] [latex]PV[/latex] [latex]i[/latex]
15. $18,302.77 $14,000.00 0.015
16. $58,870.20 $36,880 0.01

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