The logarithm of a nonpositive value is not defined.
Example Logarithmic Equation With No Solution: log(x)=log(- x) Example Logarithmic Equation With One Solution: log(x) = 10 Example Logarithmic Equation With Two Solutions: log(x^2)=2
Practice makes perfect
We are asked to write an example of a logarithmic equation with one, two, and no solutions. We will review some of the logarithmic function's characteristics that can be useful for each case, so we will work each case individually.
Example Logarithmic Equation With No Solution
When solving a logarithmic equation we have to take into account that the logarithm of a number less than or equal to zero is not defined. We can exploit this fact to write the required equation by forcing the solution to be outside of the domain.
log(x) = log(- x)
By using exponentiation in the equation above, using these common logarithms as the exponent of 10, we get to the conclusion that x=0. Let's give it a try.
The solution is x=0, but log(0) is undefined. Therefore, there is no solution for this logarithmic equation. Note that this is just an example, as there are infinitely many logarithmic equations with no solutions.
Example Logarithmic Equation With One Solution
For this case, let's start by reviewing the graph of the common logarithm function, y =log(x).
We can see that the range of the common logarithm function is all real numbers. Furthermore, this function is always increasing. This means that the function takes the value of any real number at some point and does this just once. Then, any equation with the format shown below will have just one solution.
log(x) = c
In this equation c is any real number. For example, log(x) = 10 has just one solution, as required. Nevertheless, there are infinitely many logarithmic equations having exactly one solution.
Example Logarithmic Equation With Two Solutions
For this case we need to think about a logarithmic equation such that after exponentiation it reduces to an equation with two solutions, but also ensuring that substituting it back does not lead to a negative argument. Using x^2 as argument takes care of that.
log (x^()2) = c
In this equation c can be any real number. Let's try solving an equation with this form.
Note that 10^c always gives a positive result. Then, there are always two solutions, a negative and a positive one. However, when substituting the negative solution, the argument of log (x^2) becomes positive. Therefore, the two solutions are valid. We can find an example below.
log (x^2) = 2
Notice that this is just an example, as any equations of this format have two solutions.
