One of the elements of primitive level algebra is the logarithm. The name came from Greek from the word “number” or “power” and means the power to which it is necessary to raise the number at the base to find the final number.
The logarithm of the number b to the base a is an exponent, which requires that the base a be raised to the number b. The result is pronounced like this: “logarithm of b to the base of a”. The solution to logarithmic problems is that you need to determine the given degree by the numbers by the specified numbers. There are some basic rules for determining or solving the logarithm, as well as transforming the notation itself. Using them, a solution is made logarithmic equations, derivatives are found, integrals are solved, and many other operations are performed. Basically, the solution to the logarithm itself is its simplified notation. Below are the main formulas and properties:
For any a ; a > 0; a ≠ 1 and for any x ; y > 0.
Please note: if the base logarithm is 10, then the record is shortened, a decimal logarithm is obtained. If there is a natural number e, then we write down, reducing to a natural logarithm. It means that the result of all logarithms is the power to which the base number is raised to obtain the number b.
Directly, the solution lies in the calculation of this degree. Before solving an expression with a logarithm, it must be simplified according to the rule, that is, using formulas. You can find the main identities by going back a little in the article.
Adding and subtracting logarithms with two different numbers, but with the same grounds, replace with one logarithm with the product or division of the numbers b and c, respectively. In this case, you can apply the transition formula to another base (see above).
If you are using expressions to simplify the logarithm, there are some limitations to be aware of. And that is: the base of the logarithm a is only a positive number, but not equal to one. The number b, like a, must be greater than zero.
There are cases when, having simplified the expression, you will not be able to calculate the logarithm in numerical form. It happens that such an expression does not make sense, because many degrees are irrational numbers. Under this condition, leave the power of the number as a logarithm.
For example: X 1/2 = √X.
E = lim(1+1/N), as N → ∞.
With a precision of 17 digits, the number e is 2.71828182845904512.
E(i*pi) + 1 = 0
(exp(x))" = exp(x)
Y = Logb(x).
The logarithm shows to what degree it is necessary to raise a number - the base of the logarithm (b) to get a given number (X). The logarithm function is defined for X greater than zero.
For example: Log 10 (100) = 2.
Y = Log 10 (x) .
Denoted Log(x): Log(x) = Log 10 (x).
An example of using the decimal logarithm is decibel.
Y = Log2(x).
Denoted by Lg(x): Lg(x) = Log 2 (X)
Y = loge(x) .
Denoted by Ln(x): Ln(x) = Log e (X)
The natural logarithm is the inverse of the exponential function exp(X).
Log 2 (8) = Log 10 (8) / Log 10 (2) =
0.903089986991943552 / 0.301029995663981184 = 3
Often there are problems of converting volume into area or length, and the inverse problem is converting area into volume. For example, boards are sold in cubes (cubic meters), and we need to calculate how much wall area can be sheathed with boards contained in a certain volume, see the calculation of boards, how many boards are in a cube. Or, the dimensions of the wall are known, it is necessary to calculate the number of bricks, see brick calculation.
It is allowed to use the site materials provided that an active link to the source is set.
Logarithmic Expressions, solving examples. In this article, we will consider problems related to solving logarithms. The tasks raise the question of finding the value of the expression. It should be noted that the concept of the logarithm is used in many tasks and it is extremely important to understand its meaning. As for the USE, the logarithm is used in solving equations, in applied problems, and also in tasks related to the study of functions.
Here are examples to understand the very meaning of the logarithm:
Basic logarithmic identity:
Properties of logarithms that you must always remember:
*Logarithm of the product is equal to the sum the logarithms of the factors.
* * *
* The logarithm of the quotient (fraction) is equal to the difference of the logarithms of the factors.
* * *
*Logarithm of degree is equal to the product exponent to the logarithm of its base.
* * *
*Transition to new base
* * *
More properties:
* * *
Computing logarithms is closely related to using the properties of exponents.
We list some of them:
The essence of this property is that when transferring the numerator to the denominator and vice versa, the sign of the exponent changes to the opposite. For example:
Consequence of this property:
* * *
When raising a power to a power, the base remains the same, but the exponents are multiplied.
* * *
As you can see, the very concept of the logarithm is simple. The main thing is that good practice is needed, which gives a certain skill. Certainly knowledge of formulas is obligatory. If the skill in converting elementary logarithms is not formed, then when solving simple tasks, one can easily make a mistake.
Practice, solve the simplest examples from the math course first, then move on to more complex ones. In the future, I will definitely show how the “ugly” logarithms are solved, there will be no such ones at the exam, but they are of interest, do not miss it!
That's all! Good luck to you!
Sincerely, Alexander Krutitskikh
P.S: I would be grateful if you tell about the site in social networks.
derived from its definition. And so the logarithm of the number b by reason a defined as the exponent to which a number must be raised a to get the number b(the logarithm exists only for positive numbers).
From this formulation it follows that the calculation x=log a b, is equivalent to solving the equation ax=b. For example, log 2 8 = 3 because 8 = 2 3 . The formulation of the logarithm makes it possible to justify that if b=a c, then the logarithm of the number b by reason a equals With. It is also clear that the topic of the logarithm is closely related to the topic of the power of a number.
With logarithms, as with any numbers, you can perform operations of addition, subtraction and transform in every possible way. But in view of the fact that logarithms are not quite ordinary numbers, their own special rules apply here, which are called basic properties.
Take two logarithms with the same base: log x and log a y. Then remove it is possible to perform addition and subtraction operations:
log a x+ log a y= log a (x y);
log a x - log a y = log a (x:y).
log a(x 1 . x 2 . x 3 ... x k) = log x 1 + log x 2 + log x 3 + ... + log a x k.
From quotient logarithm theorems one more property of the logarithm can be obtained. It is well known that log a 1= 0, therefore,
log a 1 /b= log a 1 - log a b= -log a b.
So there is an equality:
log a 1 / b = - log a b.
Logarithms of two mutually reciprocal numbers on the same basis will differ from each other only in sign. So:
Log 3 9= - log 3 1 / 9 ; log 5 1 / 125 = -log 5 125.
Today we will talk about logarithm formulas and give demonstration solution examples.
By themselves, they imply solution patterns according to the basic properties of logarithms. Before applying the logarithm formulas to the solution, we recall for you, first all the properties:
Now, based on these formulas (properties), we show examples of solving logarithms.
Logarithm a positive number b in base a (denoted log a b) is the exponent to which a must be raised to get b, with b > 0, a > 0, and 1.
According to the definition log a b = x, which is equivalent to a x = b, so log a a x = x.
Logarithms, examples:
log 2 8 = 3, because 2 3 = 8
log 7 49 = 2 because 7 2 = 49
log 5 1/5 = -1, because 5 -1 = 1/5
Decimal logarithm is an ordinary logarithm, the base of which is 10. Denoted as lg.
log 10 100 = 2 because 10 2 = 100
natural logarithm- also the usual logarithm logarithm, but with the base e (e \u003d 2.71828 ... - an irrational number). Referred to as ln.
It is desirable to remember the formulas or properties of logarithms, because we will need them later when solving logarithms, logarithmic equations and inequalities. Let's work through each formula again with examples.
8 2log 8 3 = (8 2log 8 3) 2 = 3 2 = 9
log 3 8.1 + log 3 10 = log 3 (8.1*10) = log 3 81 = 4
9 log 5 50 /9 log 5 2 = 9 log 5 50- log 5 2 = 9 log 5 25 = 9 2 = 81
The exponent of a logarithm number log a b m = mlog a b
Exponent of the base of the logarithm log a n b =1/n*log a b
log a n b m = m/n*log a b,
if m = n, we get log a n b n = log a b
log 4 9 = log 2 2 3 2 = log 2 3
if c = b, we get log b b = 1
then log a b = 1/log b a
log 0.8 3*log 3 1.25 = log 0.8 3*log 0.8 1.25/log 0.8 3 = log 0.8 1.25 = log 4/5 5/4 = -1
As you can see, the logarithm formulas are not as complicated as they seem. Now, having considered examples of solving logarithms, we can move on to logarithmic equations. We will consider examples of solving logarithmic equations in more detail in the article: "". Do not miss!
If you still have questions about the solution, write them in the comments to the article.
Note: decided to get an education of another class study abroad as an option.
