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Last updated on October 15, 2025
The properties of logarithms can be used to combine and simplify logarithmic expressions. Logarithms are called the inverse functions of exponents because they reverse the effects of exponentiation. Therefore, the properties of logarithms can be drawn from the properties of exponents. We can expand a logarithmic expression into multiple ones, or combine multiple logarithms into a single one. In this article, we will explore all the properties of logarithms in detail.
Each property of logarithms aims to simplify and solve logarithmic equations and expressions.
For all logarithmic properties: (m, n > 0, a > 0, a ≠ 1)
Only positive real numbers can be used to define logarithms, and their bases must be positive and not equal to 1. While more properties exist, the four basic properties are listed below:
Product property:
\( log_a mn = log_a m + log_a n \)
Condition: \((m, n > 0, a > 0, a ≠ 1)\)
This property states that the logarithm of a product \((mn)\) is equal to the sum of the logarithms of the individual factors.
Quotient property:
\(log_a m / n = log_a m - log_a n \)
Condition: \((m, n > 0, a > 0, a ≠ 1)\)
The quotient property states that the logarithm of a quotient (m/n) is the same as the difference between the logarithms of numerator and denominator.
Power property:
\(log_a(m^n) = n log_a(m)\)
Condition: \((m, n > 0, a > 0, a ≠ 1)\)
The rule shows that if we move the exponent inside a logarithm to the front of a log, then the exponent will be the multiplier.
Change of base property:
\(log_b a = log_c a / log_c b \)
Condition: \((m, n > 0, a > 0, a ≠ 1)\)
This property states that we can convert the base of a logarithm to another base.
Other properties are also directly derived from exponent rules. The definition of logarithm is:
\(a^x = m\) ⇔ \(log_a m = x\)
\(a^0 = 1\) ⇒ \(log_a 1 = 0\)
\(a^1 = a\) ⇒ \(log_a a = 1\)
\(a^{\log_a x} = x \)
\(\log_b (a^m) = m \cdot \log_b (a) \)
(Results from changing the power property and base rule).
The following table lists the properties of logarithms.
The base of the natural log is “e,” and it is expressed as loge = ln. All properties of logarithms apply to the natural log as well, with base e (ln). The natural logarithmic properties are as follows:
The product property of log explains that the logarithm of a product is equal to the sum of the two individual logs.
The property is: \(log_a mn = log_a m + log_a n \)
As we know, logarithmic forms are the inverse of the exponential forms.
For example, \(log_a m = x\) is the inverse of \(a^x = m\).
\(log_a n = y\), so \(a^y = n\)
\( mn = a^x × a^y\)
\(mn = a^{x + y}\)
\(log_a mn = x + y\)
\(y = log_a n \)
For instance, \(log(2 × 3) = log 2 + log 3\) \( = log 6 log(2 × 3) = log 6 \)
By applying the product property:
\(log_a mn = log_a m + log_a n \)
\(log(2 × 3) = log (2) + log (3)\)
Now, we can use a calculator to find the values.
\(log(2) ≈ 0.3010\)
\(log(3) ≈ 0.4771\)
Next, add 0.3010 and 0.4771.
\(0.3010 + 0.4771 = 0.7781\)
\(log(6) ≈ 0.7781\)
Therefore, \(log (2 × 3) = log 2 + log 3 = log 6\)
The logarithm of a quotient results in the difference between the logarithm of the numerator and denominator.
The quotient property of log is: \(log_a m / n = log_a m - log_a n \)
The derivation of the property is as follows:
\(m = a^x\)
\(n = a^y\)
Now, \(m / n = a^x / a^y\)
\(m /n = a^{x - y}\)
\(log_a m / n = log_a m - log_a n\)
\(log (20 / 2) = log 10 \)
\(log_a m / n = log_a m - log_a n\)
\(log (20) ≈ 1.3010\)
\(log (2) ≈ 0.3010 \)
\(log (20) - log (2) = 1.3010 - 0.3010 = 1.0000\)
\(log (10) = 1\)
Hence,\( log (20 / 2) = log (20) - log (2)\)
According to the power property, the exponent of the argument inside the logarithm can be brought to the front of another log.
\(log_a(m^n) = n log_a(m)\)
Here, the exponent n changed to the multiplier n. Let us understand the derivation of the power property.
Let \(log_a m = x\)
That is, \(a^x = m\)
Now we can raise both sides to the power of n:
\((a^x)^n = m^n\)
Now take the logarithm base a of both sides:
\(log_a (m^n) = log_a (a^{nx})\)
We assumed \(log_a (m) = x\), so:
\(log_a (m^n) = n log_a (m)\)
For instance, \(log x^3= 3 log x\)
The power property is:
\(log_a (m^n) = n log_a (m)\)
Here, base a is not given, hence it is 10 (it is a common log).
\(m = x\)
\(n = 3 \)
Now, we can apply the property:
\(log (x^3) = 3 log (x)\)
The change of base property helps us to change the base of a logarithm into another, which simplifies calculations.
The property is:
\(log_b a = log_c a / log_c b\)
According to this property, the logarithm with base b can change into any base c. It states that both logs use the same new base.
Let \(log_b a = x\), so \(a = b^x…\) (1)
Let \(log_c a = y\), so \(a = c^y …\) (2)
Let \(log_c b = z\), so \(b = c^z …\) (3)
\(zx = y ⇒ x = y / z\)
\(log_b a = log_c a / log_c b \)
\(log_b a = log_c a / log_c b\)
\(log_4 3 = (log 3) / (log 4)\)
\(log (3) ≈ 0.4771 \)
\(log (4) ≈ 0.6020 \)
\(log_4 3 = 0.4771 / 0.6020 ≈ 0.7925\)
\(4^{0.7925} ≈ 3 \)
Logarithms may seem complex at first, but once you understand their properties, solving exponential and logarithmic equations becomes simple and logical. Here are some tips and tricks to master logarithmic properties easily.
Understanding the key properties of logarithms helps students apply them when solving various mathematical problems. However, they often make some mistakes that lead to incorrect conclusions. Here are some common mistakes and their solutions to prevent them.
The properties of logarithms play an important role in various situations, from music to science and mathematics. Here are some real-world applications of the properties:
If logb 3 = a, then what is the value of log (1 / 27) in terms of a?
\(-3a \)
We are given logb 3 = a
Here, we want to find:
logb (1 / 27)
Using the quotient property of logarithms:
logb (1 / 27) = logb 1 - logb 27
Since logb 1 = 0,
logb (1 / 27) = 0 - logb 27 = - logb 27
Now, we can express 27 as a power of 3:
27 = 33
logb 27 = logb (33)
Next, apply the power property:
log (33) = 3 ⋅ logb 3 = 3a
So:
logb (1 / 27) = -3a
Hence, the answer is log (1 / 27) = -3a
Simplify log (12) = log x + log 6. Find the value of x.
x = 2
We can use the product property to find the value of x.
We can combine the two logs on the right side:
log x + log 6 = log (x 6)
Hence, the equation is:
Log 12 = log (6x)
As we know, the logarithms are equal, so their arguments must also be equal.
12 = 6x
Now, we can solve for x:
x = 12 / 6
x = 2
Therefore, the value of x is 2.
Expand log (10/2)
1 - log 2
Here, we need to expand log (10 / 2).
We can use the quotient property, which states that:
log (a / b) = log a - log b
So, the expression becomes:
log (10 / 2) = log 10 - log 2
As we know, log 10 = 1, because the base of the common logarithm is 10.
Hence, log 10 - log 2 = 1 - log 2
Therefore, the expanded expression is 1 - log 2.
Evaluate log (3^4).
log (34) = 4 log 3
We can use the power property to evaluate log (34).
It states that, loga(mn) = n loga(m)
Here, m = 3
n = 4
So, we can simplify the expressions as:
log (34) = 4 log (3)
log(81) = 4 log (3)
The simplified expression is 4 log 3.
If we want to find the approximate value of log 3, we can use a calculator.
log (3) 0.4771
Thus, 4 log 3 = 4 × 0.4771 = 1.9084
Hence, log (34) ≈ 1.9084
Evaluate using base 10: log3 9
2
The change of base property:
log3 (9) = log (9) / log (3)
Using the change of base formula, log3(9) = log(9)/log(3)
log (9) ≈ 0.9542
log (3) ≈ 0.4771
log3 (9) = 0.9542 / 0.4771 = 2
Therefore, log3 (9) = 2
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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