In this explainer, we will learn how to use natural logarithms to solve exponential and logarithmic equations.
Recall that a logarithm is simply the power a number must be raised to in order to give a certain value. For example, the logarithmic equation is another way of writing the exponential equation . We refer to as the base, as the power that is raised to (the exponent), and as the answer when we raise to the power of . To illustrate this principle, the logarithmic equation is another way of writing the exponential equation .
We should already be familiar with the properties of logarithms (often called “logs”), including the laws of logarithms, which help us to simplify expressions containing logs. We recap these laws here, as they will prove extremely useful when solving exponential and logarithmic equations.
Laws: Logarithms
For real numbers , , , and , with , the following laws of logarithms hold:
We will also need to know the following special cases:
Now, recall that the inverse of a function does the opposite to the function; for a function , we use the notation to represent its inverse. We know that the inverse of any exponential function is given by the logarithmic function . Therefore, the advantage of a logarithm is that it “undoes” exponentiation, which allows us to solve exponential equations.
In the specific case where the base is , we have the natural exponential function , with its inverse given by . We refer to as the natural logarithmic function and usually write it with the notation .
For any function with an inverse, the graph of the inverse function is a reflection of the original function in the line . Therefore, the graph of is a reflection of the graph of in the line , as shown below.
The graph of also has the following specific properties:
- It crosses the at , which tells us that .
- The is an asymptote, which shows that is defined only for positive values of and does not exist for .
- As , then too, though grows very slowly.
(Note that, for different values of , the graphs of and would look very similar to this, but with steeper or shallower curves, depending on the value of .)
Since we know that applying the inverse of a function “undoes” the action of the original function, then writing and , we have
We will use the two key facts that and throughout this explainer.
Having built up the necessary concepts, we can now tackle some equations. For instance, suppose we are asked to solve the exponential equation , giving our answer to three decimal places.
To solve for , we need a means of extracting from its position as the power (or exponent) of . Now, we already know that we can solve exponential equations such as by taking the logarithm to base 2 of both sides. We would get , which simplifies to . Therefore, since taking the natural logarithm is the inverse operation of raising to a power, here our first step is to take natural logarithms (often called “taking logs”) of both sides of the equation. Then, we simplify the result.
Taking logs and then dividing both sides by 2 gives
Using our calculator, we get . Finally, by rounding to three decimal places, we obtain the answer .
Note that if we had been asked to give our answer in exact form, we would have left the value of as the exact logarithmic expression . It is important to pay careful attention to the wording of questions to make sure that we always give our answer in its required form.
Let us now try some examples to practice these skills.
Example 1: Solving Natural Exponential Equations Using Natural Logarithms
Find, to the nearest thousandth, the value of such that .
Answer
Recall that the natural logarithmic function , usually written as , is the inverse of the natural exponential function . We can use the fact that , together with the laws of logarithms, to help us solve natural exponential equations.
Here, we have the natural exponential equation . Since taking the natural logarithm is the inverse operation of raising to a power, our first step is to take natural logarithms (often called “taking logs”) of both sides of the equation:
We solve for by adding 3 to both sides and then dividing both sides by 4, which gives
Using our calculator, we get . We were asked to round our answer to the nearest thousandth. Remember that the thousandths digit is the third digit after the decimal point, which, in this case, is a 6. The digit following this (the ten-thousandths digit) is a 1, so the answer rounds down to to the nearest thousandth.
We may be asked to solve more complex natural exponential equations. For example, suppose we have an equation such as
With questions like this, the trick is to realize that is the same as . This means that the left-hand side of the given equation is actually a quadratic function of . Hence, to solve the equation, we start by using the substitution to get
As this is now a quadratic equation in , we will factor the equation and solve for . Then, we can work backward from the fact that to find the corresponding value (or values) of .
To factor , we need to identify factor pairs that multiply to give and then select the pair that adds to give 3. It is easy to check that the required numbers are and 5, so we can factor to obtain . Thus, our equation becomes so or . That is, or .
Now, remember that for all real values of , which can be verified by referring back to our diagram showing the relationship between and . Therefore, the equation has no solution, so we are left to solve . This is now in the form of a simple natural exponential equation of the sort that we solved earlier. Taking logs of both sides gives the exact solution .
Let us test our understanding by tackling a similar type of example.
Example 2: Solving Natural Exponential Equations Using Natural Logarithms
If , where is a real number, find all possible values of .
Answer
Recall that the natural logarithmic function , usually written as , is the inverse of the natural exponential function . We can use the fact that , together with the laws of logarithms, to help us solve natural exponential equations.
In this question, we have the natural exponential equation , and we need to find all possible values of . Notice that the left-hand side is a quadratic function of , so we start by using the substitution to get
As this is now a quadratic equation in , we will factor the equation and solve for .
To factor , we need to identify factor pairs that multiply to give and then select the pair that adds to give . It is easy to check that the required numbers are and 1, so we can factor to obtain . Thus, our equation becomes so or . That is, or .
Now, remember that for all real values of . Therefore, we cannot have , so we are left with the single solution .
Next, suppose that we are asked to solve the natural logarithmic equation , giving our answer in exact form.
This time, to solve for , we need a means of extracting from its position within a natural logarithm. Since raising to a power is the inverse operation of taking the natural logarithm, our first step is to raise to the power of both sides of the equation. Then, we simplify the result by adding 1 to both sides, followed by dividing through by 3:
Our next example features a natural logarithmic equation.
Example 3: Solving Equations Involving Natural Logarithms
Solve for , giving your answer to the nearest hundredth.
Answer
Recall that the natural exponential function is the inverse of the natural logarithmic function , which is usually written as . We can use the fact that , together with the laws of logarithms, to help us solve natural logarithmic equations.
To solve the natural logarithmic equation for , we first need a means of extracting from its position as the power of 9 within the natural logarithm on the left-hand side of the equation. If we apply the power law of logarithms with , , and , we find that . Therefore, our equation becomes
Next, we divide both sides by to get
Finally, we add 9 to both sides, followed by dividing through by 7:
Using our calculator, we get . We were asked to round our answer to the nearest hundredth. Remember that the hundredths digit is the second digit after the decimal point, which in this case is a 6. The digit following this (the thousandths digit) is a 5, so the answer rounds up to to the nearest hundredth.
Sometimes, we meet exponential equations that include a combination of exponential expressions. In cases like this, we might need to apply several different operations before we get to the desired solution. Here is an example of this type.
Example 4: Solving Equations Using Natural Logarithms
Solve the equation , giving your answer in the form .
Answer
Recall that the natural logarithmic function , usually written as , is the inverse of the natural exponential function . We can use the fact that , together with the laws of logarithms, to help us solve exponential equations.
In this question, we have the exponential equation , where the left-hand side is the product of the exponential function and the natural exponential function . We know that taking the natural logarithm is the inverse operation of raising to a power. Even though only one of the left-hand exponential functions has base , it is still a sensible first step to take logs of both sides of the equation. Therefore, we get
Next, since the expression on the left-hand side is now the log of a product of two exponential functions, we can apply the multiplication law of logarithms in reverse, with , , and . This enables us to separate the two exponential functions and then simplify the second one, as follows:
Note that we can then simplify the first term on the left-hand side by applying the power law of logarithms with , , and . Thus, our equation becomes
To solve this equation for , first, we add 1 to both sides to collect the terms involving on one side and the number terms on the opposite side:
Then, we factor the left-hand side to get
Lastly, we divide through by , which gives
As we now have in the form , this is our answer.
Our final example involves a natural exponential equation taken from a real-life context.
Example 5: Solving Real-Life Problems Using Natural Logarithms
The number of people infected with measles during an outbreak in a country can be modeled by the exponential equation , where is the number of people infected after days.
- How many people were infected at the start of the outbreak?
- How many people are infected after 30 days, rounding to the nearest person?
- How many days does it take for the number of people infected to double?
Answer
Recall that the natural logarithmic function , usually written as , is the inverse of the natural exponential function . We can use the fact that , together with the laws of logarithms, to help us solve natural exponential equations.
Here, we are given the natural exponential equation , which gives the number of people infected with measles days after the start of an outbreak in a particular country.
Part 1
To work out the number of people infected at the start of the outbreak, we substitute into the equation, which gives
Part 2
To work out the number of people infected 30 days after the start of the outbreak, we substitute into the equation, which gives
As our answer represents a number of people, we must round it to the nearest whole number. Since the first digit after the decimal point is a 4, we round down to the nearest integer, which gives an answer of 907.
Part 3
To work out how many days it takes for the number of people infected to double, observe that this is equivalent to finding the value of for which is double the number infected at the start, (i.e., ). Therefore, we must solve the natural exponential equation
First, we divide both sides by 150 to get
Next, since taking the natural logarithm is the inverse operation of raising to a power, we take logs of both sides of the equation, which gives
Then, dividing through by 0.06, we have
Using our calculator, we get . As our answer represents a number of days, it must be a whole number. Since , this tells us that it takes more than 11 days for the number of people infected to double. Hence, we round up to the nearest integer, so the required number of days is 12.
Let us finish by recapping some key concepts from this explainer.
Key Points
- The natural logarithmic function is , which is usually written as . It is the inverse of the natural exponential function .
- We can use the fact that and , together with the laws of logarithms, to solve exponential and logarithmic equations.
