Get to Know the Complete Properties of Logarithms with Examples

Learn about the key properties of logarithms and why they are essential for various mathematical computations. Gain insights into their applications in science, engineering, and finance.

Properties of Logarithms

Did you know that exponents have an inverse? Yes, just like exponents which are the inverse of the power root. The inverse of exponents is logarithms. Are the properties of exponents and logarithms the same? So, what are the properties of logarithms? Rather than being curious, let’s see more!

Logarithm

Logarithm is the opposite of exponent or commonly known as exponent. The general form of logarithm can be stated as follows.

a ^b = c ⇔  ^a log c = b

With:

• a = base
• b = logarithmic result; and
• c = numeric

It turns out, there are several problems that can be solved using logarithms. For example, calculating the production of vaccines or similar products, determining the audio spectrum interval, analyzing the price of goods based on high demand and supply, and many more. You can see the complete discussion in the following article.

Properties of Logarithms

As the opposite form of exponents, are the properties of logarithms the same as the properties of exponents? Let’s take a look at some examples of the properties of logarithms below.

1. ^a log c + ^a log d = ^a log cd
2. ^a log c – ^a log d = ^a log c/d
3. ^a log c ^m = m ^a log c
4. ^a log c ^m = ^p log c / ^p log a
5. ^x log a / ^x log b = ^b log a
6. ^an log c ^m = m _d / n log c
7. a ^{a log c} = c
8. ^a log a = 1
9. ^a log 1 = 0
10. ^a log b . ^b log c = ^a log c
11. ^a log a ⁿ = n

For the properties ^a log a = 1 and ^a log 1 = 0 are called the basic properties of logarithms.

Properties of Logarithmic Operations

Logarithms can be operated like numbers, such as addition, subtraction, multiplication, and division. The discussion of each logarithmic operation is as follows.

1. Properties of Logarithmic Addition

The property of logarithmic addition is that two logarithmic numbers added together will turn into a multiplication between the numbers as long as the base is the same. This means that a logarithm can be added to another logarithm to produce a multiplication logarithm. Consider the following example.

2 log ³ + 2 log 4 = ² log(3×4)

       = ² log 12

From the example above, it is known that the logarithmic property of multiplication is a condensed form of the addition of two or more logarithms with the same base.

2. Properties of Logarithmic Subtraction

The nature of subtraction of logarithms is almost the same as addition. It’s just that two numerals in subtraction will change into division between numerals. That is, subtraction of two logarithms with the same base will produce a division logarithm. Consider the following example.

² log 8 – ² log4 = ² log (8/4) = ² log2 = 1

If we break it down one by one, is the result the same? Let’s prove it.

² log8 – ² log4 = ² log2 ³ – ² log2 ²       = 3 ² log 2 – 2 ² log2 = 3-2 = 1

From the calculation above, the same result is obtained. Quipperian can choose the method that you think is easier, but still based on the properties of logarithms, yes.

3. Properties of Logarithmic Multiplication

The multiplication property of logarithms refers to one of the 11 general properties of logarithms, namely the following properties.

^a log b. ^b log c = ^a log c

If two logarithms with different bases are multiplied, the resulting new logarithm will have the same base as the first logarithm and the same numerus as the second logarithm. Consider the following example.

³ log 2. ² log 4 = ³ log 4

Remember, logarithmic multiplication is different from logarithmic multiplication. Logarithmic multiplication is a multiplication operation between two or more logs. While log multiplication is a form of log whose numerator is a multiplication. Note the following differences.

• Log multiplication = ³ log 2 x ² log 4
• Log Multiplication = ³ log(2×4)

Therefore, the nature of both must also be different.

4. Properties of logarithmic division

The multiplication property of logarithms refers to one of the 11 general properties of logarithms, namely the following properties.

^x log a / ^x log b = ^b log a

Dividing two logarithms will produce a new log where the denominator will be the base of the new log. Meanwhile, the divisor will remain the number of the new log. Consider the following example.

2 log 4 / ² log 6 = ¹⁶ log 4

5. Properties of Roots and Square Logarithms

The properties of root and square logarithms refer to one of the 11 general properties of logarithms, namely the property ^a log c ^m = m ^a log c

This property shows that the power of a numerus can be used as a constant in front of its logarithm. Here is an example of the property of the logarithm of the quadratic form below.

⁴ log5 ² = 2 ⁴ log 5 -> properties of square logarithms

For root logarithms, you only need to change the numeric root into a power number. Next, use the properties of logarithms as explained previously. Here are some examples of the properties of root logarithms.