Proof of Logarithm Properties and Practice Problems

Proof of Logarithm Properties and Practice Problems – defantri.com

📐 Proof of Logarithm Properties and Practice Problems

The good student, learning mathematics through the proof of logarithm properties and examples. There are at least $11$ logarithm properties that we will prove in this note. The proofs below are just one alternative, so there may be other ways to prove them.

In general, a logarithm is the inverse of a power. In other words, "inverse exponential functions is called the logarithmic function".

Before we get to the properties of logarithms, let's note a little about the inverse of exponents. Let's take the simple exponential form $ {\color{Blue} 2}^{\color{Red} 3}={\color{Green} 8} $:

  • To get the number $ {\color{Blue} 2} $ using the numbers $ {\color{Red} 3} $ and $ {\color{Green} 8} $, the operation we use is the root: $ \sqrt[{3}]{{\color{Green} 8}}={\color{Blue} 2} $
  • To get the number $ {\color{Red} 3} $ using the numbers $ {\color{Blue} 2} $ and $ {\color{Green} 8} $, the operation we use is the logarithm: $ {}^{{\color{Blue} 2}}\!\log {\color{Green} 8}={\color{Red} 3} $

So the inverse of a power is not only the root but also the logarithm.
The conclusion we can draw is:

  • If $ {\color{Blue} 2}^{\color{Red} 3}={\color{Green} 8} $ then $ {}^{{\color{Blue} 2}}\!\log {\color{Green} 8}={\color{Red} 3} $
  • If $ {}^{{\color{Blue} 2}}\!\log {\color{Green} 8}={\color{Red} 3} $ then $ {\color{Blue} 2}^{\color{Red} 3}={\color{Green} 8} $

In logical terms, we can write:
$ {\color{Blue} 2}^{3}={\color{Green} 8} $ if and only if $ {}^{{\color{Blue} 2}}\!\log {\color{Green} 8}={\color{Red} 3} $

The general form of a logarithm can be written as follows:
$ {}^{{\color{Blue} a}}\!\log {\color{Green} b}={\color{Red} c} $ if and only if $ {\color{Blue} a}^{\color{Red} c}={\color{Green} b} $

Note: The notation $ {}^{{\color{Blue} a}}\!\log {\color{Green} b}={\color{Red} c} $ is commonly found in Indonesian textbooks, while in international books (English), the notation is $ \log_{{\color{Blue} a}}{\color{Green} b}={\color{Red} c} $.

The form $ {}^{{\color{Blue} a}}\!\log {\color{Green} b}={\color{Red} c} $ is read as: "the logarithm of $ {\color{Green} b} $ with base $ {\color{Blue} a} $ is $ {\color{Red} c} $." For convenience, it is often simply called "a log b = c".
Terminology of Logarithm $ {}^{{\color{Blue} a}}\!\log {\color{Green} b}={\color{Red} c} $:
  • $ {\color{Blue} a} $ is called the base. Conditions: $ {\color{Blue} a} > 0 $ and $ {\color{Blue} a} \neq 1 $. For base $ 10 $, it may be omitted.
  • $ {\color{Green} b} $ is called the numerus (the number whose logarithm is sought). Condition: $ {\color{Green} b} > 0 $
  • $ {\color{Red} c} $ is called the result of the logarithm

After knowing the general form of logarithms above, let's explore some logarithm properties.

📖 List of Logarithm Properties
1. $ {}^{a}\!\log a = 1 $
2. $ {}^{a}\!\log 1 = 0 $
3. $ {}^{a}\!\log x + {}^{a}\!\log y = {}^{a}\!\log (x \cdot y) $
4. $ {}^{a}\!\log x - {}^{a}\!\log y = {}^{a}\!\log \dfrac{x}{y} $
5. $ {}^{a}\!\log x^{n} = n \cdot {}^{a}\!\log x $
6. $ {}^{a^{n}}\!\log x^{m} = \dfrac{m}{n} \cdot {}^{a}\!\log x $
7. $ {}^{a}\!\log x = \dfrac{{}^{p}\!\log x}{{}^{p}\!\log a} $
8. $ {}^{a}\!\log x = \dfrac{1}{{}^{x}\!\log a} $
9. $ {}^{a}\!\log x \cdot {}^{x}\!\log b = {}^{a}\!\log b $
10. $ a^{{}^{a}\!\log x} = x $
11. $ a^{{}^{b}\!\log c} = c^{{}^{b}\!\log a} $

By getting to know the logarithm properties above, we can get closer to logarithms. When we are close to logarithms, we will easily know when a logarithm can or cannot be used to solve a problem.

1. Proof of $ {}^{a}\!\log a = 1 $
🔍 Alternative Proof
To prove this logarithm property, let's assume we don't know the value of $ {}^{a}\!\log a $, so we can set it as a variable $ y $: \[ {}^{a}\!\log a = y \] Using the general form of logarithms $ {}^{{\color{Blue} a}}\!\log {\color{Green} b}={\color{Red} c} \iff {\color{Blue} a}^{\color{Red} c}={\color{Green} b} $: \[ {\color{Blue} a}^{\color{Red} y} = {\color{Green} a} \implies {\color{Blue} a}^{\color{Red} y} = {\color{Green} a}^{\color{Red} 1} \] For the equation to hold, $ y = 1 $, so: \[ {}^{a}\!\log a = 1 \] Thus, we have proven that $ {}^{a}\!\log a = 1 $.
Examples:
  • $ {}^{2}\!\log 2 = 1 $
  • $ {}^{5}\!\log 5 = 1 $
  • $ \log 10 = 1 $
2. Proof of $ {}^{a}\!\log 1 = 0 $
🔍 Alternative Proof
Let $ {}^{a}\!\log 1 = y $. Using the general form: \[ {}^{a}\!\log 1 = y \iff a^y = 1 \] Since $ a^0 = 1 $, we get $ y = 0 $: \[ {}^{a}\!\log 1 = 0 \] Thus, we have proven that $ {}^{a}\!\log 1 = 0 $.
Examples:
  • $ {}^{2}\!\log 1 = 0 $
  • $ {}^{5}\!\log 1 = 0 $
  • $ \log 1 = 0 $
3. Proof of $ {}^{a}\!\log x + {}^{a}\!\log y = {}^{a}\!\log (x \cdot y) $
🔍 Alternative Proof
Let $ {}^{a}\!\log x = m $ and $ {}^{a}\!\log y = n $. Then: \[ a^m = x, \quad a^n = y \] Multiplying: \[ x \cdot y = a^m \cdot a^n = a^{m+n} \] Using the general form: \[ {}^{a}\!\log (x \cdot y) = m + n = {}^{a}\!\log x + {}^{a}\!\log y \] Thus, we have proven that $ {}^{a}\!\log x + {}^{a}\!\log y = {}^{a}\!\log (x \cdot y) $.
Examples:
  • $ {}^{2}\!\log 3 + {}^{2}\!\log 5 = {}^{2}\!\log 15 $
  • $ {}^{7}\!\log 3 + {}^{7}\!\log 10 = {}^{7}\!\log 30 $
  • $ {}^{5}\!\log 2 + {}^{5}\!\log 3 + {}^{5}\!\log 4 = {}^{5}\!\log 24 $
4. Proof of $ {}^{a}\!\log x - {}^{a}\!\log y = {}^{a}\!\log \dfrac{x}{y} $
🔍 Alternative Proof
Let $ {}^{a}\!\log x = m $ and $ {}^{a}\!\log y = n $. Then $ a^m = x $ and $ a^n = y $: \[ \frac{x}{y} = \frac{a^m}{a^n} = a^{m-n} \] Thus: \[ {}^{a}\!\log \left( \frac{x}{y} \right) = m - n = {}^{a}\!\log x - {}^{a}\!\log y \] Thus, we have proven that $ {}^{a}\!\log x - {}^{a}\!\log y = {}^{a}\!\log \dfrac{x}{y} $.
Examples:
  • $ {}^{2}\!\log 3 - {}^{2}\!\log 5 = {}^{2}\!\log \frac{3}{5} $
  • $ {}^{5}\!\log 10 - {}^{5}\!\log 2 = {}^{5}\!\log 5 = 1 $
  • $ {}^{3}\!\log 16 - {}^{3}\!\log 4 - {}^{3}\!\log 2 = {}^{3}\!\log 2 $
5. Proof of $ {}^{a}\!\log x^{n} = n \cdot {}^{a}\!\log x $
🔍 Alternative Proof
Using the property of exponents on $ x^n $, we get multiplication of $ x $ repeated $ n $ times: \[ {}^{a}\!\log x^n = {}^{a}\!\log (x \cdot x \cdot x \cdots x) \] Using the previous property (3): \[ {}^{a}\!\log x^n = {}^{a}\!\log x + {}^{a}\!\log x + \cdots + {}^{a}\!\log x = n \cdot {}^{a}\!\log x \] Thus, we have proven that $ {}^{a}\!\log x^{n} = n \cdot {}^{a}\!\log x $.
Examples:
  • $ {}^{2}\!\log 64 = {}^{2}\!\log 2^6 = 6 \cdot {}^{2}\!\log 2 = 6 $
  • $ \log 1000 = \log 10^3 = 3 $
  • $ {}^{5}\!\log \frac{1}{5} = {}^{5}\!\log 5^{-1} = -1 $
6. Proof of $ {}^{a^{n}}\!\log x^{m} = \dfrac{m}{n} \cdot {}^{a}\!\log x $
🔍 Alternative Proof
Let $ {}^{a^{n}}\!\log x^{m} = y $. Then: \[ (a^n)^y = x^m \implies a^{ny} = x^m \] Taking $ {}^{a}\!\log $ on both sides: \[ ny \cdot {}^{a}\!\log a = m \cdot {}^{a}\!\log x \implies ny = m \cdot {}^{a}\!\log x \] Thus: \[ y = \frac{m}{n} \cdot {}^{a}\!\log x \] Thus, we have proven that $ {}^{a^{n}}\!\log x^{m} = \dfrac{m}{n} \cdot {}^{a}\!\log x $.
Examples:
  • $ {}^{4}\!\log 125 = {}^{2^2}\!\log 5^3 = \frac{3}{2} \cdot {}^{2}\!\log 5 $
  • $ {}^{27}\!\log 8 = {}^{3^3}\!\log 2^3 = {}^{3}\!\log 2 $
  • $ {}^{25}\!\log 0.001 = {}^{5^2}\!\log 10^{-3} = -\frac{3}{2} \cdot {}^{5}\!\log 10 $
7. Proof of $ {}^{a}\!\log x = \dfrac{{}^{p}\!\log x}{{}^{p}\!\log a} $
🔍 Alternative Proof
Let $ {}^{a}\!\log x = y $. Then $ a^y = x $. Taking $ {}^{p}\!\log $ on both sides: \[ {}^{p}\!\log a^y = {}^{p}\!\log x \implies y \cdot {}^{p}\!\log a = {}^{p}\!\log x \] Thus: \[ y = \frac{{}^{p}\!\log x}{{}^{p}\!\log a} \] Thus, we have proven that $ {}^{a}\!\log x = \dfrac{{}^{p}\!\log x}{{}^{p}\!\log a} $.
Examples:
  • $ {}^{2}\!\log 3 = \dfrac{{}^{5}\!\log 3}{{}^{5}\!\log 2} $
  • If $ {}^{2}\!\log 5 = m $, then $ {}^{5}\!\log 2 = \dfrac{1}{m} $
  • If $ {}^{2}\!\log 3 = n $, then $ {}^{27}\!\log 16 = \dfrac{4}{3n} $
8. Proof of $ {}^{a}\!\log x = \dfrac{1}{{}^{x}\!\log a} $
🔍 Alternative Proof
Using property 7 with $ p = x $: \[ {}^{a}\!\log x = \frac{{}^{x}\!\log x}{{}^{x}\!\log a} = \frac{1}{{}^{x}\!\log a} \] Thus, we have proven that $ {}^{a}\!\log x = \dfrac{1}{{}^{x}\!\log a} $.
Examples:
  • $ {}^{2}\!\log 3 = \dfrac{1}{{}^{3}\!\log 2} $
  • If $ {}^{3}\!\log 5 = p $, then $ {}^{5}\!\log 3 = \dfrac{1}{p} $
  • If $ {}^{2}\!\log 3 = m $, then $ {}^{27}\!\log 16 = \dfrac{4}{3m} $
9. Proof of $ {}^{a}\!\log x \cdot {}^{x}\!\log b = {}^{a}\!\log b $
🔍 Alternative Proof
Using property 7 with $ p = 10 $ (or any base): \[ {}^{a}\!\log x \cdot {}^{x}\!\log b = \frac{\log x}{\log a} \cdot \frac{\log b}{\log x} = \frac{\log b}{\log a} = {}^{a}\!\log b \] Thus, we have proven that $ {}^{a}\!\log x \cdot {}^{x}\!\log b = {}^{a}\!\log b $.
Examples:
  • $ {}^{2}\!\log 3 \cdot {}^{3}\!\log 5 = {}^{2}\!\log 5 $
  • $ {}^{3}\!\log 2 \cdot {}^{5}\!\log 27 \cdot {}^{2}\!\log 5 = 3 $
  • $ {}^{3}\!\log 2 \cdot {}^{8}\!\log 81 = \dfrac{4}{3} $
10. Proof of $ a^{{}^{a}\!\log x} = x $
🔍 Alternative Proof
Let $ {}^{a}\!\log x = y $. Then by definition: \[ a^y = x \] Substituting $ y = {}^{a}\!\log x $: \[ a^{{}^{a}\!\log x} = x \] Thus, we have proven that $ a^{{}^{a}\!\log x} = x $.
Examples:
  • $ 3^{{}^{3}\!\log 2} = 2 $
  • $ 32^{{}^{2}\!\log 5} = 5^4 $
  • $ 3^{{}^{27}\!\log 2} = 2^{\frac{1}{3}} $
11. Proof of $ a^{{}^{b}\!\log c} = c^{{}^{b}\!\log a} $
🔍 Alternative Proof
Let $ y = a^{{}^{b}\!\log c} $. Taking $ \log $ on both sides: \[ \log y = {}^{b}\!\log c \cdot \log a = \frac{\log c}{\log b} \cdot \log a \] Rearranging: \[ \log y = \frac{\log a}{\log b} \cdot \log c = {}^{b}\!\log a \cdot \log c = \log c^{{}^{b}\!\log a} \] Thus: \[ y = c^{{}^{b}\!\log a} \] Thus, we have proven that $ a^{{}^{b}\!\log c} = c^{{}^{b}\!\log a} $.
Examples:
  • $ 2^{{}^{3}\!\log 5} = 5^{{}^{3}\!\log 2} $
  • $ 10^{{}^{7}\!\log 3} = 3^{{}^{7}\!\log 10} $
  • $ 10^{\log 15} = 15 $

As a practice exercise in using these logarithm properties, please check out the logarithm problems and solutions that have been tested in School Exams or University Entrance Tests.

💡 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.

Note: This document on Proof of Logarithm Properties is a "living document" that is always being improved or updated according to changing needs. Any additional input from you to the admin is expected to improve the quality of this note. 🙏 CMIIW

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