Lesson 21 MA 152, Sections 4.3 and 4.4
Since an exponential function is 1-1, it has an inverse. Remember the generalizations about the graph of an exponential function?
- The y-intercept is (0, 1).
- The graph is increasing if b > 1, decreasing if 0 < b < 1.
- The range is all positive numbers, so graph lies above the x-axis.
- The x-axis is an asymptote.
- The point (1, b) is on the graph.
The inverse of this function, , has a graph with the following characteristics.
- The x-intercept is (1, 0).
- The graph still is increasing if b > 1, decreasing if 0 < b < 1.
- The domain is all positive numbers, so the graph is to the right of the y-axis. (The range is all real numbers.)
- The y-axis is an asymptote.
- The point (b, 1) is on the graph.
Below is a graph of .
If you imagine the line y = x, you can see the symmetry about that line.
Below are both graphs on the same coordinate system along with y = x.
The inverse of an Exponential Function is called a Logarithmic Function.
If , the logarithmic function with base b is defined by
. The domain of this function is and the range is . It is denoted as .
A logarithmic function can be expressed in exponential form or logarithmic form. It is necessary for you to be able to convert between these two forms.
Exponential Form Logarithmic Form
argument base exponent exponent base argument
Ex 1: Convert each exponential form to logarithmic form and each logarithmic form to exponential form.
The exponent y in either form is called a logarithm.
Ex 2: Find each logarithm.
Ex 3: Find the value of a in each logarithmic equation.
If the base of a logarithmic equation is 10, the exponent is called a common logarithm. If there is no base indicated, assume the base is 10.
Ex 4: Find each common logarithm.
A calculator can be used to solve some logarithmic equations when the base is 10. The key is the common logarithmic key. If you enter a number and press that key,
the answer is the exponent that used with base 10 equals that number. For example, if 12 is used, the log is 1.0792 (to 4 decimal places). That means . 1.0792 is the solution of . You are finding the exponent or the logarithm.
If you want to solve , you can use the 2nd key with the log key. That will give the 12th power of 10. You are finding the power of 10, the argument.
Likewise a calculator can be used to solve some logarithmic equations when the base is e. The key is the natural logarithm key. If you want to find the value of ln 12,
it is 2.4849 (to 4 decimal places). That means . 2.4849 is the solution of You are finding the exponent or logarithm.
If you want to solve , you can use the 2nd key with the ln key. That will give the 12th power of e. You are finding the power of e, the argument.
Ex 5: Use a calculator to find a in each of the following.
Ex 6: Solve each equation. Round to 4 decimal places, if necessary.
Ex 7:
Ex 8: Find the value of x in the following.
Just as there are several real world applications that involve formulas with exponential expressions, there are several real world applications that involve logarithmic expressions.
The measure of voltage gain of devices such as amplifiers or the length of a transmission line is called a decibel. If is the output voltage of a device and is the input voltage, the decibel voltage gain is given by .
Ex 9: Find the db gain of an amplifier whose input voltage is 0.71 volt and whose output voltage is 20 volts.
The measurement of the intensity of an earthquake is a number from the Richter Scale. If R represents the intensity, A is the amplitude (measured in micrometers), and P is the period (time of one oscillation of the Earth's surface, measured in seconds), then . (Notice the Richter scale is based on common logarithms. An earthquake of 4.0 is 10 times greater than one of 3.0.)
Ex 10: Find the intensity of an earthquake with an amplitude of 6000 micrometers and a period of 0.08 second. Round to the nearest tenth.
The time in minutes required to charge a battery depends on how close it is to being fully charged. If M is the theoretical maximum charge, k is a positive constant that is dependent upon the type of battery and charger, C is the given level of M to which the battery is being charged, then the time t required to reach that level is given by . Note: C is usually a percentage of M.
Ex 11: If k = 0.201, how long will it take a battery to reach a 40% charge? Assume that the battery was fully discharged when it began charging.
If a population is growing exponentially at a certain annual rate, then the time required for that population to double is called the doubling time and is given by , where t is time in years.
Ex 12: A town's population is growing at 9.2% per year. If this growth rate remains constant, how long will it take for the town's population to double?
When energy is added to a gas and its temperature remains constant, the volume could increase. This is called isothermal expansion. If the temperature remains at T (in Kelvins), the energy (in joules) required to increase the volume of 1 mole of gas from an initial volume Vi to a final volume Vf is , where R is the universal gas constant, which is 8.314 joules/mole/K.
Ex 13: Find the amount of energy that must be supplied to triple the volume of 1 mole of gas at a constant temperature of 350K.
If an investment is growing continuously for t years, its annual growth rate r is given by the formula , where P is the current value and P0 is the initial value of the investment.
Ex 14: An investment of $10,400 in America Online in 1992 was worth $10,400,000 in 1999. Find AOL's average annual growth rate during this period.
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