Learn the essential rules of logarithms and elevate your mathematical skills. Discover how to simplify expressions and solve for exponents effortlessly.
Let’s figure out what a logarithm is, what its types and properties are, and also find out the expert’s opinion on why the ability to calculate logarithms is needed at all
From ancient Greek, “logarithm” is translated as “number of relations”. The first to talk about logarithms was the Scottish mathematician John Napier in 1614. Then logarithmic tables were actively used in the work of scientists and engineers, but the need for them disappeared with the advent of calculators and computers. However, today’s schoolchildren should be able to solve logarithmic problems and examples.
What is logarithm in algebra
A logarithm is an exponent to which the base must be raised to obtain a certain number. At the same time, the logarithm has a condition that the base must be greater than zero and not equal to one, and the resulting number must be greater than zero. Also, the logarithm can have an integer part – the characteristic, and a fractional part – the mantissa.
Useful information about logarithms
| The logarithm is defined only for positive numbers. | This is due to the fact that the exponential function can only take positive values. |
| What is the logarithm for? | Logarithms make it easier to compare quantities that differ by several times. |
| There is a concept of a “logarithmic spiral” | It is along this line that the horns of mountain goats grow, many galaxies are twisted, the shells of some sea creatures, the tendrils of plants, hurricanes, and tornadoes |
| You need to be careful when writing the logarithm. | Already at this stage, some students make mistakes, confusing the base with the degree. |
Base of logarithm
The base of the logarithm is considered its main part. If several logarithmic functions have the same base, then various operations can be performed with them. The base of the natural logarithm is called Euler’s number – this is a number approximately equal to 2.71828.
There are several requirements for the base of the logarithm:
- it must be greater than zero;
- it cannot be equal to one. Because no matter to what power one is raised, the result will always be one.
Rules of Logarithms:
Logarithms are in the nature of exponents; therefore, they obey certain rules closely related to the rules of exponents. These can be of great help in simplifying mathematical operations The first three rules are stated only in terms of natural log, but they are also valid when the symbol in is replaced by logb.
Rule 1 (log of product) In (uv) = 1n + 1n v (u, v >0) Example 1 lnl(e6e4) = Ine6 +lne4 =6 + 4 = 10
Example 2 ln(Ae7) = InA + Ine’ = In A+ 7
Rule II log of quotient)
ln(p/u) = In u – In o (p,u>0)
Examp!e3 ln(e2/c) = Ine2-Inc = 2-lnc
Example 4 ln(e2 /e5) = Ine2 – Ine5 = 2-5 = -3
Rule III (log of a power) In p° = a in p (p > 0)
Examples: Ine15 =151ne = 15
Examples:. In A3 =3 In A
Example 7 ln(pua) = lnp + lnualnp + alnu
Rule IV (conversion of log base)
logb u = (logb e)(logc p) (p > 0)
This rule, which resembles the chain rule in spirit (witness the “chain” enables us to derive a
logarithm • be loge M (to bae )e) from the logarithm logb M (to bse b), or vice versa.
Rule IV can readily be generalized to
logbu = (l°gb C)(l°gc P)
where c is some base other than b.
Types of logarithms
There are several main types of logarithms. Let’s take a closer look at them.
Natural logarithm
The natural logarithm is the logarithm to the base a, where a is a number equal to approximately 2.72. The natural logarithm is denoted in writing as ln b, log a b, or sometimes simply log b if the base a is implied.
Example:
ln a = 1; a = a 1
Natural logarithms are used in solving algebraic equations where the unknown is an exponent, and they are also necessary in mathematical analysis.
Decimal logarithm
A decimal logarithm is a logarithm whose base is 10. The decimal logarithm of a number a is written as: log(a) or lg(a). That is, the decimal logarithm of a number a is the solution to the equation 10 x = a.
In this case, a distinction is made between real and complex decimal logarithms. The real decimal logarithm of a number a will exist only if a is greater than zero, while the complex logarithm will exist if a is not equal to zero.
The common logarithm is useful when working with round numbers.
Binary logarithm
The binary logarithm is the logarithm to the base 2. To find the binary logarithm of a number a, it is necessary to solve the equation 2 x = a.
The binary logarithm of a number exists if that number is greater than zero.
The binary logarithm is denoted as lb a, lb(a), log 2 a.
Example:
lb 1 = 0.
The fundamental logarithmic property
The fundamental logarithmic identity follows from the definition of the logarithm:
x log x Y = y
It follows that the equality of two real logarithms implies the equality of the expressions being logarithmed. That is, if log x y = log x z, then x logx Y = x logx Z . Then, according to the fundamental identity, y = z.
Other properties of logarithms
Knowing the properties of logarithms will help to reduce the logarithm of a complex expression to simple arithmetic operations on logarithms.
- The logarithm of one to any base is zero: log x 1 = 0.
- The logarithm of a product is equal to the sum of the logarithms of the factors: log x AB = log x A + log x B.
- The logarithm of the quotient is equal to the difference of the logarithms: log x A / B = log x A – log x B.
- The logarithm of a power is equal to the logarithm of the modulus of the base multiplied by the exponent: log x (A b ) = b × log x A.
- The logarithm of the root is equal to the logarithm of the modulus of the expression under the radical divided by the root factor: log x a √B = log x B / a.
- The relationship between logarithms with different bases is determined by the formula: log x A = 1 / log y X × log y A.
Problems and examples on the topic “Logarithm”
Let’s consolidate the theory into practice and solve several examples on the topic.
Example 1
Solve the equation: log 5 (5 – x) = 2 log 5 3
Example 2
log 2 (4 – x) = 7
Task 1
36 log6 5
Answers to problems and examples
Now let’s check the solutions and compare the answers we get.
Solution 1
5 – x = 3 2
5 – x = 9
X = -4
Solution 2
log 2 (4 – x) = log 2 2 7
4 – x = 2 7
4 – x = 128
X = -124