7.2 Logarithmic Functions
If a location has an R number of 1.2 for coronavirus and they have 1500 hospital beds available. Assuming they are starting with 2000 active cases today and about 20% of coronavirus patients will need a hospital bed, how many weeks will it be before their hospitals are overwhelmed? Remember the exponential function for this was . The hospitals can handle 1500 patients so we need to know the number of cases the location can handle. 20% of that number is 1500. So:
So the location will be overwhelmed after 7500 active cases, to find the number of weeks this will be, we solve the following equation:
We can divide both sides by 2000:
While we have set up exponential models and used them to make predictions, you may have noticed that solving exponential equations has not yet been mentioned. The reason is simple: none of the algebraic tools discussed so far are sufficient to solve exponential equations. Consider the equation above. We can use our calculator to determine that and , so it is clear that x must be some value between 7 and 8 since the function is increasing. We could use technology to create a table of values or graph to better estimate the solution.
Note that the graph of intersects with at , so we know that the number of weeks to 7500 cases and thus a requirement of 1500 or more hospital beds is a little over 7 weeks away, which might cause local officials to create mitigating efforts to lower the R number.
This result is still fairly unsatisfactory, since it would be nice to have a function that gives the answer. None of the functions we have already discussed would work, so we must introduce a new function, named log, as the function that “undoes” an exponential function, like how a square root “undoes” a square. Since exponential functions have different bases, we will define corresponding logarithms of different bases as well.
Logarithm
The logarithm (base b) function, written , “undoes” exponential function .
The statement is equivalent to the statement .
Since the logarithm and exponential “undo” each other (in technical terms, they are inverses), the following properties of logs are no surprise.
Properties of Logs: Inverse Properties
Since log is a function, it is most correctly written as , using parentheses to denote function evaluation, just as we would with . However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written as .
Example Using Inverse Properties of Logs
a. Write these exponential equations as logarithmic equations:
i.
This is equivalent to
ii.
This is equivalent to
iii.
This is equivalent to
b. Write these logarithmic equations as exponential equations:
i.
This is equivalent to
ii.
This is equivalent to
Try it Now 1
a. Write the exponential equation as a logarithmic equation.
b. Write the logarithmic equation
By establishing the relationship between exponential and logarithmic functions, we can now solve basic logarithmic and e