Properties of Logarithms

The goal of each property of logarithms is 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: 

    loga mn = loga m + loga 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 two logarithms. 

Quotient property:

     loga m / n = loga m - loga 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 two logarithms. 

Power property:

   loga( mn) = n loga 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:

   logb a = logc a / logc 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: 
       ax = m  loga m = x

a0 = 1  loga 1 = 0 

a1 = a  loga a = 1

aloga x = x 

logbn  (am) = m ⋅ logb (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. Here, all the above properties are in the “log” form, so those are also applicable to the natural log. The natural logarithmic properties are as follows:

ln (1) = 0 

Any number raised to the power of 0 is equal to 1. 

ln (e) = 1

The natural log of e is always 1 because e raised to the power of 1 is equal to e itself. 

ln (mn) = ln m + ln n 

The natural logarithm of the product (mn) is equal to the sum of the natural logarithms of m and n. 

ln (m / n) = ln m - ln n 
The difference between the natural logs of m and n is equal to the division of m and n. 

ln (mn) = n × ln (m) 

When a number m is raised to a power n, the exponent n can be moved in front of the logarithm as a multiplier. 

e(ln (x)) = x 

When e is raised to the power of ln (x), the result will be x itself. Both ln and e cancel out, and we get back the number x. 
 

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:
     loga mn = loga m + loga n 

As we know, logarithmic forms are the inverse of the exponential forms. 
For example, loga m = x is the inverse of ax = m.

Here, loga m = x, so ax = m 
loga n = y, so ay = n 

Now, we can multiply mn as:  
   mn = ax × ay

By using the multiplying exponents, 
    mn = ax + y

Now, let us convert it to logarithmic form:
    loga mn = x + y 

Hence, x = loga m 
y = loga n 

Therefore, loga mn = loga m + loga n 

For instance, log(2 × 3) = log 2 + log 3  = log 6
  log(2 × 3) = log 6 

By applying the product property: 
     loga mn = loga m + loga 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:
        loga m / n = loga m - loga n 

The derivation of the property is as follows:
 
Let loga m = x and loga n = y
 The exponential forms are: 
   m = ax
    n = ay 

Now, m / n = ax / ay

By using the rule of dividing exponents, 
    m /n = ax - y 

Next, convert it to logarithmic form:
     loga m / n = loga m - loga n 

For example, log (20 / 2) = log 20 - log 2  
    log (20 / 2) = log 10 

By applying the quotient property: 
    loga m / n = loga m - loga n 

We can use a calculator to find the value of log (20) and log (2). 
   log (20)  1.3010
   log (2)  0.3010 

Now, we can subtract:
    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. 
  
     loga( mn) = n loga m 
 
Here, the exponent n changed to the multiplier n. Let us understand the derivation of the power property. 

Let loga m = x
That is, ax = m 

Now we can raise both sides to the power of n:
   (ax)n = mn

Now take the logarithm base a of both sides:
    loga (mn) = loga (anx)

We assumed loga (m) = x, so: 
 
    loga (mn) = n loga (m)

For instance, log x3 = 3 log x 
The power property is:
     loga (mn) = n loga (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 (x3) = 3 log (x)
 

The change of base property helps us to change the base of a logarithm into another, which makes the calculations easier. The property is:
    
    logb a = logc a / logc 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.  

The derivation of the change of base property is:

Let logb a = x, so a = bx… (1)
Let logc a = y, so a = cy … (2) 
Let logc b = z, so b = cz … (3)
    
Now, we can replace b in the first equation with cz.
Then, a = (cz)x = czx
However, from previously, a = cy

Hence, czx = cy
Now that the bases are the same, we can equate the exponents:
 zx = y   x = y / z

Finally, substituting back: 
          logb a = logc a / logc b  

For example, log4 3 = (log 3) / (log 4) 
We can apply the property:
       logb a = logc a / logc b 
Next, substitute the values:
      log4 3 = (log 3) / (log 4)

Now, we can use a calculator to find the values. 
log (3)  0.4771 
log (4)  0.6020 

Here, we divide the two values:
   log4 3 = 0.4771 / 0.6020  0.7925 
If we raise 4 to the power of 0.7925:
 40.7925  3  
 

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:
 

• Seismologists can use the properties of logarithms to measure the magnitude of earthquakes. For example, to compare the magnitudes, they can use the power and properties of logs, which are measured on a logarithmic scale. 
  
   
• Scientists can use the properties of logarithms to measure the alkalinity or acidity of various solutions in their laboratories. For instance, the acidity of a lemon solution is measured using the pH scale, then, 
   pH = -log10[H+] 
  pH = - log (0.001) = -(-3) = 3
  This means that the pH of lemon juice is 3, which makes it acidic. 
  
   
• To make long-term predictions and plans related to population, officials can use the properties of logarithms. This is because the properties help them to simplify complex population models.