📐 Properties of Logarithms

Sahabat Smaridasa — To be able to solve logarithm problems, the most important thing to master is the properties of logarithms. Most logarithm problems that appear in the National Examination or University Entrance Tests are solved using the properties of logarithms. Let's take a look at the properties below.

Properties of Logarithms are basic material that we must master thoroughly and we must know how to use them. These logarithm properties can be thought of as tools to calculate and determine the result of a logarithmic form. Without understanding the properties of logarithms, it will be difficult for us to solve problems directly related to logarithms.

To make it easier to remember the Properties of Logarithms, we need to practice many logarithm problems with various types of questions. If necessary, we should work on problems from university entrance tests, because those problems are usually very challenging. By regularly working on logarithm problems, we will indirectly remember the properties automatically.

📖 Properties of Logarithms

For $ a > 0, a \neq 1, b > 0, c > 0 $, the following properties of logarithms apply:

(i) $ {}^a \log 1 = 0 $
(ii) $ {}^a \log a = 1 $
(iii) $ {}^a \log (b \cdot c) = {}^a \log b + {}^a \log c $
(iv) $ {}^a \log \frac{b}{c} = {}^a \log b - {}^a \log c $
(v) $ a^{{}^a \log b} = b $
(vi) $ {}^{a^m} \log b^n = \frac{n}{m} \cdot {}^a \log b $
       1) $ {}^{a^m} \log b = \frac{1}{m} \cdot {}^a \log b $
       2) $ {}^a \log b^n = n \cdot {}^a \log b $
       3) $ {}^{a^m} \log b^n = {}^a \log b^{\frac{n}{m}} $
       4) $ {}^{a^m} \log b^n = {}^{a^{\frac{m}{n}}} \log b $
(vii) $ {}^a \log b = \frac{{}^p \log b}{{}^p \log a} $, with $ p > 0, p \neq 1 $
       1) $ {}^a \log b = \frac{1}{{}^b \log a} $
       2) $ {}^a \log b \cdot {}^b \log c = {}^a \log c $

Below are some examples of the logarithm properties mentioned above.

Example 1

Determine the value of $ {}^5 \log 1 $ and $ {}^7 \log 7 $.

Solution:

* $ {}^5 \log 1 = 0 $, because $ 5^0 = 1 $
* $ {}^7 \log 7 = 1 $, because $ 7^1 = 7 $

Example 2

If $ \log 2 = 0.301 $ and $ \log 3 = 0.477 $, determine the value of $ \log 6 $.

Solution:

Based on property (iii):
$ \log 6 = \log (2 \cdot 3) = \log 2 + \log 3 = 0.301 + 0.477 = 0.778 $

Answer: $ \log 6 = 0.778 $

Example 3

If $ \log 2 = 0.301 $, determine the value of $ \log 5 $.

Solution:

Based on property (iv):
$ \log 5 = \log \frac{10}{2} = \log 10 - \log 2 = 1 - 0.301 = 0.699 $

Answer: $ \log 5 = 0.699 $

Example 4

Determine the value of $ 3^{{}^3 \log 7} $.

Solution:

Based on property (v):
$ 3^{{}^3 \log 7} = 7 $

Answer: $ 3^{{}^3 \log 7} = 7 $

Example 5

Determine the value of $ {}^\sqrt{2} \log 8 $.

Solution:

Based on property (vi):
$ {}^\sqrt{2} \log 8 = {}^{2^{\frac{1}{2}}} \log 2^3 = \frac{3}{\frac{1}{2}} \cdot {}^2 \log 2 = 6 \cdot 1 = 6 $

Answer: $ {}^\sqrt{2} \log 8 = 6 $

Example 6

Determine the value of $ {}^5 \log 625 $.

Solution:

Based on property (vi):
$ {}^5 \log 625 = {}^5 \log 5^4 = 4 \cdot {}^5 \log 5 = 4 \cdot 1 = 4 $

Answer: $ {}^5 \log 625 = 4 $

Example 7

Determine the value of $ {}^2 \log 3 \cdot {}^\sqrt{3} \log 16 $.

Solution:

Based on properties (vi) and (vii):
\[ \begin{aligned} {}^2 \log 3 \cdot {}^\sqrt{3} \log 16 &= {}^2 \log 3 \cdot {}^{3^{\frac{1}{2}}} \log 2^4 \\ &= {}^2 \log 3 \cdot \frac{4}{\frac{1}{2}} \cdot {}^3 \log 2 \\ &= 8 \cdot {}^2 \log 3 \cdot {}^3 \log 2 \\ &= 8 \cdot {}^2 \log 2 \\ &= 8 \cdot 1 = 8 \end{aligned} \] Answer: $ {}^2 \log 3 \cdot {}^\sqrt{3} \log 16 = 8 $

Example 8

If $ {}^2 \log 3 = p $ and $ {}^2 \log 5 = q $, then express the following logarithms in terms of $ p $ and $ q $:
a) $ {}^2 \log 15 $    b) $ {}^{12} \log 20 $

Solution:

a) Based on property (iii):
$ {}^2 \log 15 = {}^2 \log (3 \cdot 5) = {}^2 \log 3 + {}^2 \log 5 = p + q $

b) Based on properties (vii) and (iii):
\[ \begin{aligned} {}^{12} \log 20 &= \frac{{}^2 \log 20}{{}^2 \log 12} \\ &= \frac{{}^2 \log (4 \cdot 5)}{{}^2 \log (4 \cdot 3)} \\ &= \frac{{}^2 \log 4 + {}^2 \log 5}{{}^2 \log 4 + {}^2 \log 3} \\ &= \frac{2 + q}{2 + p} \end{aligned} \] Answers:
a) $ {}^2 \log 15 = p + q $
b) $ {}^{12} \log 20 = \frac{2 + q}{2 + p} $

Actually, to solve logarithm problems, the properties used are free from properties (i) to (vii). If the logarithm properties used are appropriate, the solution will be shorter. However, if the properties used are not appropriate, the solution will take longer, but you can be sure that the answer will be found.

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💡 Tip: The best way to master logarithm properties is through continuous practice. Try various problems involving different combinations of logarithm properties until you can recognize which property to use in each situation.