The ratio of natural logarithms. Logarithmic expressions. examples

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What is a logarithm?

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What is a logarithm? How to solve logarithms? These questions confuse many graduates. Traditionally, the topic of logarithms is considered complex, incomprehensible and scary. Especially - equations with logarithms.

This is absolutely not true. Absolutely! Don't believe? Good. Now, for some 10 - 20 minutes you:

1. Understand what is a logarithm.

2. Learn to solve a whole class of exponential equations. Even if you haven't heard of them.

3. Learn to calculate simple logarithms.

Moreover, for this you will only need to know the multiplication table, and how a number is raised to a power ...

I feel you doubt ... Well, keep time! Go!

First, solve the following equation in your mind:

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

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 logarithm is closely related to the topic degree of number.

With logarithms, as with any numbers, you can perform addition operations, 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.

Addition and subtraction of logarithms.

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.