There are a few logarithmic laws worth knowing to make solving problems easier. The following laws are the main rules you will use. It should be noted that the letters a, m, n must be positive real numbers for these laws to be valid.
There are a few logarithmic laws worth knowing to make solving problems easier. The following laws are the main rules you will use. It should be noted that the letters a, m, n must be positive real numbers for these laws to be valid.
\( \log_{10}3+\log_{10}4= \)
Constant Values:
It can be automatically determined that:
Basic Arithmetic Operations
Multiplication, division, subtraction, and addition operations between logarithms:
Changing the Base of a Logarithm:
It should be noted that on calculators the default value is a logarithm based on . But sometimes we want to calculate a logarithm with a base other than . For example, what if we want to know to what power we must raise to get the number ? (The answer is, of course, , because raised to the power of equals ). To perform this change, the base change of the algorithm must be used. There are two ways to do this:
Derivative of the Logarithm:
Integral of the Logarithm:
There's a main method for calculating logarithms in high school, based on the definition of the log term. As mentioned, log is actually an inverse operation to a normal power. Therefore, if we want to find out what is, we have to ask ourselves, "What power of three would be equal to nine?" And we'll find that the answer is two. In other words, we can mark the answer with an and create an equation:
We can say that:
And create a simple exponential equation for the solution.
In more complicated exercises, previous logarithmic laws should be used to simplify the solution.
For example: take the logarithm
According to the Logarithmic Law we can say that:
And so we can know that
With a bit of practice, you'll see that solving logarithms isn't as scary or daunting as it seems, and like everything in math, practice and understanding are the name of the game!
\( \log_24+\log_25= \)
\( \log_974+\log_9\frac{1}{2}= \)
\( \log_53-\log_52= \)
First, to understand what logarithms are, one must understand what an exponential function is. An exponential function is one of the most useful functions in the language of mathematics and expresses interesting growth and decay processes. Due to the great importance of the exponential function, it is one of the important topics in the study of mathematics. An exponential function is actually a function of the type , where a is the base of the function.
A well-known exponential function is one based on the number e, which is a mathematical constant equal to the value . This function is special for several reasons, one of which is that its derivative is equal to itself.
A logarithmic function is the opposite of an exponential function, and it actually answers the question: "To what power do we need to raise a given number to obtain another given number?" For example, if I want to know to what power I need to raise to get the number , on the calculator, I would enter: log. The result will be , since raised to the power of is .
So far we have only seen what a common logarithm is, but there is another type of logarithm, which is known as the natural logarithm, and this logarithm is the inverse function of the exponential. That is, the natural logarithm has a base , this base is an irrational number which is a constant and its value is:
And the natural logarithm can be represented as:
Properties of the natural logarithm
Although the logarithm does not have a base as such, it is understood that its base is , so we must raise to the power of zero to get the result .
We can see this as:
These two properties are because they are inverse operations.
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\( \log_29-\log_23= \)
\( \log_75-\log_72= \)
\( \log_49\times\log_{13}7= \)
Assignment
is equal to . What is the whole number?
Solution
To find the whole number, we multiply a fractional number by and divide by the percentage number percentages
Answer
\( \log_mn\times\log_zr= \)
\( 2\log_38= \)
\( 3\log_76= \)
Assignment
Solution
Where:
and
Therefore
Answer
Task
Solution
Answer
\( \frac{\log_85}{\log_89}= \)
\( \frac{1}{\log_49}= \)
\( 2\log_82+\log_83= \)
Assignment
Solution
Let's break it down into parts
We substitute back into the equation
Answer
Assignment
Solution
Let's break it down into parts
Answer
\( \frac{1}{2}\log_39-\log_31.5= \)
\( \frac{1}{5}\log_81024-2\log_8\frac{1}{2}= \)
\( \log_{10}3+\log_{10}4= \)
Assignment
Find
Solution
Multiply by:
Extract the square root
Answer
Assignment
How much is ?
Solution
Definition domain
We reduce by: and by
Not in the definition domain
Definition domain
Answer
\( \log_24+\log_25= \)
\( \log_974+\log_9\frac{1}{2}= \)
\( \log_53-\log_52= \)
What are the elements of a logarithm?
The logarithm function has four elements:
Where:
is its symbol
is the argument of the logarithm
is the base of the logarithm
is the result of the logarithm
It reads: "The logarithm base of equals ”
What is the relationship between the exponential function and the logarithmic function and some examples?
A logarithm is the inverse operation of the exponential function, it is important to remember that all operations have an inverse operation, in this case, every logarithmic function can be related to an exponential one.
Example 1:
Calculate
We must find a number which will be the exponent that the base must have to obtain the logarithm's argument, which is , in this case the result will be , since
Result
Example 2:
Calculate
Now we must find the exponent that the base must have to obtain as a result the argument of the logarithm, in this case the result is , because:
Result
From these examples, we can conclude the very close relationship that exists between the logarithmic function and the exponential function.
What is a logarithm used for?
A logarithm is used to determine what number the base of the logarithm must be raised to in order to obtain the argument.
What is a logarithm and examples?
As we have already said, a logarithm is the inverse operation of an exponential. Therefore, the logarithmic function is the one where we must calculate the exponent's number that the base must have so that it gives us the number of the argument; here we will present some examples:
Examples:
Since,
As
This is because:
In this last example, the logarithm does not have an explicit base, but when a logarithm does not have a base as such, we must assume that it is a base logarithm, therefore the answer to this logarithm is , since .
What are the laws of logarithms, examples?
Let's look at some of the laws of logarithms:
It should be mentioned that there are no logarithms of negative numbers, nor of negative bases, and neither does the logarithm of zero exist.
Examples where we can apply these laws
Example 1:
And this law is too obvious since , therefore the exponent must be , to obtain the argument of the logarithmic function. And that is why law mentioned above means that if you have a logarithm with the same base and argument, the result will always be .
Example 2:
Find the value of , in the following logarithmic equation
We can use the sum of logarithms and obtain:
We use the third law mentioned above:
We have:
From here we solve for
Therefore, the value of
And we can verify:
, here we substitute the value we have found for
and as we well know:
and
, then;
Result
Another way to solve this exercise is as follows:
From here we can calculate the second logarithm:
, then we substitute this value
From here we solve for the logarithm:
Then here we already have the exponent that the base must have, which should be , therefore , and from here we obtain the argument of the algorithm, that is the value of
Or using law we have
Result
What is the difference between a common logarithm and a natural logarithm?
The difference is only the base, as the common logarithm has base : and the natural logarithm has base
Although in reality both logarithms do not have their bases implicit, each of the bases should be understood.
\( \log_29-\log_23= \)
\( \log_75-\log_72= \)
\( \log_49\times\log_{13}7= \)