In this section, we introduce a new mathematical tool called a logarithm, which will help us solve exponential equations. Much the same way as a square root "undoes" the squaring function, a logarithm undoes an exponential function.
We answer questions of this type by writing and solving an exponential equation. The initial value of the population is , and the growth factor is . Thus, the function
gives the number of bacteria present on day , and we must solve the equation
We are looking for an unknown exponent on base 2. Dividing both sides by 50 yields
This equation asks the question: "To what power must we raise 2 in order to get 16?"
This equivalence tells us that the logarithm, , is the same as the exponent in . We see again that a logarithm is an exponent; it is the exponent to which must be raised to yield .
For example, to convert the equation to exponential form, we note that the base is and the logarithm is , so the exponent on base 5 will be 3, like this: .
The conversion equations allow us to convert from logarithmic to exponential form, or vice versa. It will help you to become familiar with the conversion, because we will use it frequently.
We use logarithms to solve exponential equations, just as we use square roots to solve quadratic equations. Consider the two equations
and
We solve the first equation by taking a square root, and we solve the second equation by computing a logarithm:
and
The operation of taking a base logarithm is the inverse operation for raising the base to a power, just as extracting square roots is the inverse of squaring a number.
Now let’s consider computing logarithms that are not obvious by inspection. Suppose we would like to solve the equation
The solution of this equation is , but can we find a decimal approximation for this value? There is no integer power of 2 that equals 26, because
and
So must be between 4 and 5. We can use trial and error to find the value of to the nearest tenth. Use your calculator to make a table of values for , starting with and using increments of 0.1.
We graph ^ X and in the same window to obtain the graph shown below. Next we activate the intersect feature to find that the two graphs intersect at the point . Because this point lies on the graph of , we know that
Some logarithms are used so frequently in applications that their values are programmed into scientific and graphing calculators. These are the base 10 logarithms, such as
and
Base 10 logarithms are called common logarithms, and the subscript 10 is often omitted, so that is understood to mean .
To evaluate a base 10 logarithm, we use the LOG key on a calculator. Many logarithms are irrational numbers, and the calculator gives as many digits as its display allows. We can then round off to the desired accuracy.
We can check the approximations found in Example 7.3.15 with our conversion equations. Remember that a logarithm is an exponent, and in this example the base is 10. We find that
and
so our approximations are reasonable, although you can see that rounding a logarithm to 2 decimal places does lose some accuracy.
Do not omit the parenthesis when entering the expression in Example 7.3.19. Without the parenthesis, you are calculating . You can check that this is not the same as .
We have seen that exponential functions are used to describe some applications of growth and decay, . There are two common questions that arise in connection with exponential models:
Given a value of , what is the corresponding value of ?
To answer the first question, we evaluate the function at the appropriate value. To answer the second question, we must solve an exponential equation, and this usually involves logarithms.
At this stage, it seems we will only be able to solve exponential equations in which the base is 10. However, we will see in future sections how the properties of logarithms enable us to solve exponential equations with any base.
Radium, a radioactive element, was used to paint watches and instrument dials until its serious health effects were discovered in the 1920’s. Although some isotopes of radium take hundres of years to decay, Radium-228 decays according to the formula
The atmospheric pressure at the top of Mount McKinley, the highest peak in the United States, is 13.51 inches of mercury. Estimate the elevation of Mount McKinley.