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Section0.4Logarithmic Functions

Objectives
  • How can we “undo” the effects of exponentiation?

  • How can we solve equations involving exponential and logarithmic expressions?

  • What are the properties of logarithmic functions?

In section 2 we studied exponential functions to model a variety of different settings. It is straightforward to verify that the graph of an exponential function passes the “horizontal line test” described in section 3, and so we should expect exponential functions to have corresponding inverse functions. In this section we will define the logarithm to be the inverse function for an exponential.

Preview Activity0.4.1

Carbon-14 (\(^{14}\)C) is a radioactive isotope of carbon that occurs naturally in the Earth's atmosphere. During photosynthesis, plants take in \(^{14}\)C along with other carbon isotopes, and the levels of \(^{14}\)C in living plants are roughly the same as atmospheric levels. Once a plant dies, it no longer takes in any additional \(^{14}\)C. Since \(^{14}\)C in the dead plant decays at a predictable rate (the half-life of \(^{14}\)C is approximately 5,730 years), we can measure \(^{14}\)C levels in dead plant matter to get an estimate on how long ago the plant died. Suppose that a plant has 0.02 milligrams of \(^{14}\)C when it dies.

  1. Write a function that represents the amount of \(^{14}\)C remaining in the plant after \(t\) years.

  2. Complete the table for the amount of \(^{14}\)C remaining \(t\) years after the death of the plant.

    t 0 1 5 10 100 1000 2000 5730
    \(^{14}\)C Level 0.02

  3. Suppose our plant died sometime in the past. If we find that there are 0.014 milligrams of \(^{14}\)C present in the plant now, estimate the age of the plant to within 50 years.

Subsection0.4.1Logarithms

The Logarithm

Let \(b>0\) with \(b\neq1\text{.}\) The logarithm of \(x\) with base \(b\) is defined by

\begin{equation*} \log_{b}x=y\qquad\mbox{if and only if} \qquad x=b^{y}. \end{equation*}

The expression \(\log_{b}x\) represents the power to which \(b\) needs to be raised in order to get \(x\text{.}\)

Two frequently used logarithmic functions are \(\log_{10}x\) (frequently written \(\log x\) without the base explicitly stated) and the natural logarithm \(\log_{e}x\) (frequently written \(\ln x\)).

Note that we have specifically defined logarithms to be inverse functions for exponentials. For instance, \(\log_{10}{1000} = 3\text{,}\) since \(10^{3} = 1000\text{.}\) Logarithmic functions give us a way to re-write exponential expressions and, more importantly, solve equations involving variables in an exponent.

Subsection0.4.2Properties of Logarithms

Since logarithms and exponentials are inverse functions, many of the properties of logarithmic functions can be deduced directly from the properties of exponential functions. For example, the domain of all logarithmic functions is \((0,\infty)\) and the range of all logarithmic functions is \((-\infty,\infty)\) because those are the range and domain, respectively, of exponential functions. Similarly, logarithmic functions have a vertical asymptote at \(x = 0\) because exponential functions have a horizontal asymptote at \(y = 0\text{.}\) These features can be seen in Figure 2.

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Figure0.4.2Graphs of the functions \(y=e^{x}\) and \(y=\ln{x}\text{.}\)

The following properties of logarithms can be deduced from the properties of exponential functions and the definition of the logarithm. These properties are especially useful in simplifying or solving logarithmic and exponential equations.

Properties of Logarithms

For \(b>0\text{,}\) \(b \ne 1\text{,}\) and \(x,y>0\text{:}\)

  1. \(\log_{b}1=0\) (which is the same as \(b^0 = 1\))

  2. \(\log_{b}b=1\) (which is the same as \(b^1 = b\))

  3. \(\log_{b}\left(xy\right)=\log_{b}x+\log_{b}y\) (which is the same as \(b^mb^n=b^{m+n}\))

  4. \(\log_{b}\left(\dfrac{x}{y}\right)=\log_{b}x-\log_{b}y\) (which is the same as \( \frac{b^m}{b^n} = b^{m-n}\))

  5. \(\log_{b}x^{r}=r\,\log_{b}x\) (which is the same as \((b^m)^n = b^{mn}\))

  6. \(\log_{b}b^{x}=x\)

  7. \(b^{\log_{b}x}=x\)

  8. \(\log_{b}x=\log_{b}y\) if and only if \(x=y\)

  9. \(\log_a b = \frac{\log_c b}{\log_c a}\) (this is called the change of base formula)

Euler's number, \(e\text{,}\) shows up so often that we give a special name to the associated logarithm. The logarithm \(\log_e (x)\) is called the natural logarithm and is written as \(\ln(x)\text{.}\)

The Natural Logarithm

The natural logarithm is the logarithm with Euler's number, \(e\), as the base. \begin{equation*} \log_{e}(x) = \ln(x) \end{equation*} Note that \( \ln(a) = b \) is the same as \( e^b = a \).

We will see later in this course that the exponential function \(f(x) = e^x\) has very special calculus properties and as such the natural logarithm has very special properties as well. With this in mind, most mathematicians and scientists use \(e^x\) and \(\ln(x)\) as their preferred exponential and logarithmic functions.

Activity0.4.2

Use the definition of a logarithm along with the properties of logarithms to answer the following.

  1. Write the exponential expression \(8^{1/3} = 2\) as a logarithmic expression.

  2. Write the logarithmic expression \(\log_2 \frac{1}{32} = -5\) as an exponential expression.

  3. What value of \(x\) solves the equation \(\log_2 x = 3\)?

  4. What value of \(x\) solves the equation \(\log_2 4 = x\)?

  5. Use the laws of logarithms to rewrite the expression \(\log \left( x^3 y^5 \right)\) in a form with no logarithms of products, quotients, or powers.

  6. Use the laws of logarithms to rewrite the expression \(\log \left( \frac{x^{15} y^{20}}{z^4} \right)\) in a form with no logarithms of products, quotients, or powers.

  7. Rewrite the expression \(\ln(8) + 5 \ln(x) + 15 \ln(x^2+8)\) as a single logarithm.

Hint
Solution
Example 0.4.4

Find a value of \(x\) for which \(3^{x}=13\text{.}\)

Solution
Example 0.4.5

In 1970, the population of the United States was approximately 205.1 million people. Since that time, the population has grown at a continuous growth rate of approximately 1.05%. Assuming that this growth rate continues, when would we expect the population of the United States to reach 350 million?

Solution
Activity0.4.3

Solve each of the following equations for \(t\), and verify your answers using a calculator.

  1. \(\ln t=4\)

  2. \(\ln(t+3)=4\)

  3. \(\ln(t+3)=\ln4\)

  4. \(\ln(t+3)+\ln(t)=\ln4\)

  5. \(e^{t}=4\)

  6. \(e^{t+3}=4\)

  7. \(2e^{t+3}=4\)

  8. \(2e^{3t+2}=3e^{t-1}\)

Hint
Solution
Activity0.4.4

Consider the following equation: \begin{equation*} 7^{x} = 24 \end{equation*}

  1. How many solutions should we expect to find for this equation?

  2. Solve the equation using the log base 7.

  3. Solve the equation using the log base 10.

  4. Solve the equation using the natural log.

  5. Most calculators have buttons for \(\log_{10}\) and \(\ln\), but none have a button for \(\log_{7}\). Use your previous answers to write a formula for \(\log_{7}x\) in terms of \(\log x\) or \(\ln x\).

Hint
Solution
Activity0.4.5
  1. In the presence of sufficient resources the population of a colony of bacteria exhibits exponential growth, doubling once every three hours. What is the corresponding continuous (percentage) growth rate?

  2. A hot bowl of soup is served at a dinner party. It starts to cool according to Newton's Law of Cooling so its temperature, \(T\) (measured in degrees Fahrenheit) after \(t\) minutes is given by \begin{equation*} T(t) = 65 + 186 e^{-0.06t}. \end{equation*} How long will it take from the time the food is served until the temperature is \(120^\circ\)F?

  3. The velocity (in ft/sec) of a sky diver \(t\) seconds after jumping is given by \begin{equation*} v(t) = 80\left( 1-e^{-0.2t} \right). \end{equation*} After how many seconds is the velocity 75 ft/sec?

Hint
Solution

Subsection0.4.3Summary

  • A logarithmic function can be written in the form \(f(x)=\log_{b}x\) where \(b>0\text{,}\) \(b\ne1\text{,}\) and \(x>0\text{.}\)

  • Logarithmic functions are defined to be inverse functions for exponentials. That is

    \begin{equation*} \log_{b}x=y\qquad\mbox{if and only if} \qquad x=b^{y}. \end{equation*}
  • Solving equations that contain exponential expressions frequently requires the use of logarithms; solving equations that contain logarithmic expressions frequently requires the use of exponentials.

Subsubsection0.4.4Exercises

1

Use the laws of logarithms to rewrite the expression

\begin{equation*} \ln \left( x^{13} \sqrt{\frac{y^9}{z^2} } \right) \end{equation*}

in a form with no logarithm of a power, product, or quotient.

2

Solve \(\ln(5x^2+2) = 4\) for \(x\text{.}\) Give only an exact answer (no decimal approximations).

3

A wooden artifact from an ancient tomb consists of 20% of the carbon-14 that is present in living trees. How long ago was the artifact made assuming that the half-life of carbon-14 is 5,730 years?