7.5 Common and Natural Logarithms

Learning Outcome

  • Evaluate expressions or functions of the form $$e^x$$, $$10^x$$, $$\ln{x}$$, or $$\log{x}$$ using a scientific calculator.
  • Find the domain and range of functions of the form $$f(x)=e^x$$, $$f(x)=10^x$$, $$f(x)=\ln{x}$$, or $$f(x)=\log{x}$$ using their graphs and write them in interval notation.
  • Match the graph of a function of the form $$f(x)=e^x$$, $$f(x)=10^x$$, $$f(x)=\ln{x}$$, or $$f(x)=\log{x}$$ with the corresponding equation.

As we saw earlier, the amount earned on an account increases as the compounding frequency increases. The table below shows that the increase from annual to semi-annual compounding is larger than the increase from monthly to daily compounding. This might lead us to ask whether this pattern will continue.

Examine the value of [latex]$1[/latex] invested at [latex]100\%[/latex] interest for [latex]1[/latex] year, compounded at various frequencies.

Frequency [latex]A\left(t\right)={\left(1+\frac{1}{n}\right)}^{n}[/latex] Value
Annually [latex]{\left(1+\frac{1}{1}\right)}^{1}[/latex] [latex]$2[/latex]
Semiannually [latex]{\left(1+\frac{1}{2}\right)}^{2}[/latex] [latex]$2.25[/latex]
Quarterly [latex]{\left(1+\frac{1}{4}\right)}^{4}[/latex] [latex]$2.441406[/latex]
Monthly [latex]{\left(1+\frac{1}{12}\right)}^{12}[/latex] [latex]$2.613035[/latex]
Daily [latex]{\left(1+\frac{1}{365}\right)}^{365}[/latex] [latex]$2.714567[/latex]
Hourly [latex]{\left(1+\frac{1}{\text{8766}}\right)}^{\text{8766}}[/latex] [latex]$2.718127[/latex]
Once per minute [latex]{\left(1+\frac{1}{\text{525960}}\right)}^{\text{525960}}[/latex] [latex]$2.718279[/latex]
Once per second [latex]{\left(1+\frac{1}{31557600}\right)}^{31557600}[/latex] [latex]$2.718282[/latex]

These values appear to be approaching a limit as n increases. In fact, as n gets larger and larger, the expression [latex]{\left(1+\frac{1}{n}\right)}^{n}[/latex] approaches a number used so frequently in mathematics that it has its own name: the letter [latex]e[/latex]. This value is an irrational number, which means that its decimal expansion goes on forever without repeating. Its approximation to six decimal places is shown below.

A General Note: The Number [latex]e[/latex]

The letter e represents the irrational number

[latex]{\left(1+\frac{1}{n}\right)}^{n}[/latex]
as n increases without bound

The letter e is used as a base for many real-world exponential models. To work with base e, we use the approximation, [latex]e\approx 2.718282[/latex]. The constant was named by the Swiss mathematician Leonhard Euler (1707–1783) who first investigated and discovered many of its properties.

In our first example, we will use a calculator to find powers of e.

Example

Calculate [latex]{e}^{3.14}[/latex]. Round to five decimal places.

Natural Logarithms

The most frequently used base for logarithms is e. Base e logarithms are important in calculus and some scientific applications; they are called natural logarithms. The base e logarithm, [latex]{\mathrm{log}}_{e}\left(x\right)[/latex], has its own notation, [latex]\mathrm{ln}\left(x\right)[/latex].

Most values of [latex]\mathrm{ln}\left(x\right)[/latex] can be found only using a calculator. The major exception is that, because the logarithm of [latex]1[/latex] is always [latex]0[/latex] in any base, [latex]\mathrm{ln}1=0[/latex]. For other natural logarithms, we can use the [latex]\mathrm{ln}[/latex] key that can be found on most scientific calculators. We can also find the natural logarithm of any power of e using the inverse property of logarithms.

A General Note: Definition of the Natural Logarithm

A natural logarithm is a logarithm with base e. We write [latex]{\mathrm{log}}_{e}\left(x\right)[/latex] simply as [latex]\mathrm{ln}\left(x\right)[/latex]. The natural logarithm of a positive number x satisfies the following definition.

For [latex]x>0[/latex],

[latex]y=\mathrm{ln}\left(x\right)\text{ can be written as }{e}^{y}=x[/latex]

We read [latex]\mathrm{ln}\left(x\right)[/latex] as, “the logarithm with base e of x” or “the natural logarithm of x.”

The logarithm y is the exponent to which e must be raised to get x.

Since the functions [latex]y=e{}^{x}[/latex] and [latex]y=\mathrm{ln}\left(x\right)[/latex] are inverse functions, [latex]\mathrm{ln}\left({e}^{x}\right)=x[/latex] for all x and [latex]e{}^{\mathrm{ln}\left(x\right)}=x[/latex] for x > [latex]0[/latex].

In the next example, we will evaluate a natural logarithm using a calculator.

Example

Evaluate [latex]y=\mathrm{ln}\left(500\right)[/latex] to four decimal places using a calculator.

In our next video, we show more examples of how to evaluate natural logarithms using a calculator.

Common Logarithms

Sometimes we may see a logarithm written without a base. In this case, we assume that the base is [latex]10[/latex]. In other words, the expression [latex]{\mathrm{log}}_{}[/latex] means [latex]{\mathrm{log}}_{10}[/latex]. We call a base-[latex]10[/latex] logarithm a common logarithm. Common logarithms are used to measure the Richter Scale mentioned at the beginning of the section. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.

Definition of Common Logarithm: Log is an exponent

A common logarithm is a logarithm with base [latex]10[/latex]. We write [latex]{\mathrm{log}}_{10}(x)[/latex]  simply as [latex]{\mathrm{log}}_{}(x)[/latex].  The common logarithm of a positive number, x, satisfies the following definition:

For [latex]x\gt0[/latex],

[latex]y={\mathrm{log}}_{}(x)[/latex] can be written as [latex]10^y=x[/latex]

We read [latex]{\mathrm{log}}_{}(x)[/latex] as ” the logarithm with base [latex]10[/latex] of x” or “log base [latex]10[/latex] of x”.

The logarithm y is the exponent to which 10 must be raised to get x.

Example

Evaluate [latex]{\mathrm{log}}_{}(1000)[/latex] without using a calculator.

Example

Evaluate [latex]y={\mathrm{log}}_{}(321)[/latex] to four decimal places using a calculator.

In our last example, we will use a logarithm to find the difference in magnitude of two different earthquakes.

Example

The amount of energy released from one earthquake was [latex]500[/latex] times greater than the amount of energy released from another. The equation [latex]10^x=500[/latex] represents this situation, where x is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

Summary

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally because the logarithm is an exponent. Logarithms most commonly use base 10 and natural logarithms use base e. Logarithms can also be evaluated with most kinds of calculator.