How to solve logarithmic equations?
Everyone knows why mathematics is needed. However, many people need help in solving mathematical problems and equations. Before telling you how to solve logarithmic equations, you need to understand what they are. Equations that contain the unknown at the base of the logarithm or under its sign are called logarithmic equations. An equation having the form: logaX = b, or those that can be reduced to this form, is considered to be the simplest logarithmic equations.
For the correct solution of such equations, it is necessary to remember the properties of any logarithmic function:
- set of real numbers (range)
- set of positive numbers (domain)
- in the case when "a" is greater than 1, there is a strict increase in the logarithmic function, if less - decrease
- with the given parameters: loga "a" equals 1, and also loga 1 equals zero, you need to take into account that "a" will not be equal to 1, and will be greater than 0.
The correct solution of logarithmic equations depends on the understanding of the logarithm itself. Take an example: 5x = 11. X is a number in which you need to build 5 to get 11. This number is called the logarithm of 11 on the base 5 and it is written in the following form: x = log511. Thus, we managed to solve the exponential equation: 5x = 11, having received the answer: x = log511.
An equation that has logarithms is called logarithmic equations. In this equation, the unknown variables, as well as expressions with them, are located inside the logarithms themselves. And nowhere else! Examples of logarithmic equations: log2x = 16, log5 (x3-7) = log5 (3x), lg3 (x + 3) + 20 = 15lg (x + 5), etc. Do not forget that various expressions with x-mi can be found only inside a given lagorithm.
Getting rid of logarithms
Methods for solving logarithmic equations are applied in accordance with the existing problem, and the process of solving as a whole is a very difficult task. Let's start with elementary equations. The simplest logarithmic equations are as follows:
- logx-21 = 11
- log5 (70x-1) = 2
- log5x = 25
The solution of the logarithmic equation involves a transition from an equation with logarithms, to an equation in which there are none. And in the simplest equations it can be done in one step. It is for this reason that they are called the simplest.For example, we need to solve the following equation: log5x = log52. For this we do not need special knowledge. In this example, we need to get rid of logarithms, which spoil the whole picture. Remove the logarithms and we get: x = 2. Thus, in the future it is necessary to remove unnecessary logarithms, if possible. After all, just such a sequence allows solving logarithmic inequalities and equations. In mathematics, such actions are called potentiation. But such a deliverance from logarithms has its own rules. If the logarithms do not have any coefficients (i.e., are given by themselves), and also with their identical numerical basis, the logarithms can be removed.
Remember, after we have eliminated logarithms, we still have a simplified equation. Let's solve another example:
log9 (5x-4) -log9x. Potentiate and we get:
- 5x-4 = x
- 5x = x + 4
- 4x = 4
- x = 1
As you can see, removing the logarithms, we got the usual equation, which is no longer difficult to solve. Now you can go to more complex examples: log9 (60x-1) = 2. We need to refer to the logarithm (the number in which the base is raised, in our case, 9) to get the sub-logarithm expression (60x-1). Our logarithm is 2.Therefore: 92 = 60x-1. Logarithm is no more. Solve the resulting equation: 60x-1 = 59, x = 1.
This example we decided according to the meaning of the logarithm. It should be noted that from any given number you can make a logarithm, and the required form. This method is very useful in solving inequalities and logarithmic equations. If you need to find the root in the equation, let's look at how this can be done: log5 (18 - x) = log55
If in our equation both sides of the equation have logarithms having the same basis, then we can equate the expressions that stand under the signs of our logarithms. We remove the common base: log5. We get a simple equation: 18 –x = 5, x = 13.
In fact, solving logarithmic equations is not so difficult. Even taking into account the fact that the properties of logarithmic equations may differ significantly, all the same - there are no unsolvable tasks. It is necessary to know the properties of the logarithm itself, as well as be able to apply them correctly. No need to hurry: we recall the above instructions and proceed to the solution of the tasks. In no case do not need to be afraid of a complex equation, you have all the necessary knowledge and resources to easily cope with any of them.