# Nakafa Framework: LLM

URL: /en/subject/high-school/10/mathematics/exponential-logarithm/logarithm-properties
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/mathematics/exponential-logarithm/logarithm-properties/en.mdx

Output docs content for large language models.

---

export const metadata = {
  title: "Logarithm Properties",
  description: "Discover 8 essential logarithm properties: product, quotient, power rules with proofs. Master change of base formula and solve complex equations.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/04/2025",
  subject: "Exponents and Logarithms",
};

## Basic Properties of Logarithms

Like exponents, logarithms also have several important properties that need to be understood. These properties will be very helpful in solving various logarithmic problems.

Let <InlineMath math="a > 0" /> and <InlineMath math="a \neq 1" />, <InlineMath math="b > 0" />, <InlineMath math="c > 0" />, <InlineMath math="m > 0" />, <InlineMath math="m \neq 1" />, where <InlineMath math="a, b, c, m, n" /> are real numbers <InlineMath math="(a, b, c, m, n \in \mathbb{R})" />. The following are **logarithm properties**:

1. <InlineMath math="^a\log a = 1" />
2. <InlineMath math="^a\log 1 = 0" />
3. <InlineMath math="^a\log a^n = n" />
4. <InlineMath math="^a\log (b \times c) = ^a\log b + ^a\log c" />
5. <InlineMath math="^a\log \left(\frac{b}{c}\right) = ^a\log b - ^a\log c" />
6. <InlineMath math="^a\log b^n = n \cdot ^a\log b" />
7. <InlineMath math="^a\log b = \frac{^m\log b}{^m\log a} = \frac{1}{^b\log a}" />
8. <InlineMath math="^a\log b \times ^b\log c = ^a\log c" />

## Proving Logarithm Properties

### Logarithm of Multiplication

**Property 4**: <InlineMath math="^a\log (b \times c) = ^a\log b + ^a\log c" />

Proof: Let <InlineMath math="^a\log b = m" /> and <InlineMath math="^a\log c = n" />

This means:

<InlineMath math="b = a^m" /> and <InlineMath math="c = a^n" />

Using the property of exponents:

<BlockMath math="b \times c = a^m \times a^n = a^{m+n}" />

Therefore:

<BlockMath math="^a\log (b \times c) = ^a\log (a^{m+n}) = m + n = ^a\log b + ^a\log c" />

### Logarithm of Division

**Property 5**: <InlineMath math="^a\log \left(\frac{b}{c}\right) = ^a\log b - ^a\log c" />

Proof: Let <InlineMath math="^a\log b = m" /> and <InlineMath math="^a\log c = n" />

Then <InlineMath math="b = a^m" /> and <InlineMath math="c = a^n" />

Recall that <InlineMath math="\frac{a^m}{a^n} = a^{m-n}" />, so:

<BlockMath math="\frac{b}{c} = \frac{a^m}{a^n} = a^{m-n}" />

Therefore:

<BlockMath math="^a\log \left(\frac{b}{c}\right) = ^a\log (a^{m-n}) = m - n = ^a\log b - ^a\log c" />

### Logarithm of Power

**Property 6**: <InlineMath math="^a\log b^n = n \cdot ^a\log b" />

Proof: Let <InlineMath math="^a\log b = m" />

<InlineMath math="^a\log b^n" /> means the logarithm of <InlineMath math="b" /> raised
to the power of <InlineMath math="n" />

<InlineMath math="^a\log b^n = ^a\log (\underbrace{b \times b \times b \times \ldots \times b}_{n \text{ factors}})" />

Using property 4 repeatedly:

<BlockMath math="^a\log b^n = \underbrace{^a\log b + ^a\log b + ^a\log b + \ldots + ^a\log b}_{n \text{ factors}} = n \cdot ^a\log b" />

### Change of Base

**Property 7**: <InlineMath math="^a\log b = \frac{^m\log b}{^m\log a} = \frac{1}{^b\log a}" />

Proof:
Based on the definition of logarithm, <InlineMath math="^a\log b = c" /> if and only if <InlineMath math="b = a^c" />

Suppose we use base <InlineMath math="m" /> for the logarithm of <InlineMath math="b" />:

<BlockMath math="^m\log b = ^m\log a^c" />

Using property 6:

<BlockMath math="^m\log b = c \cdot ^m\log a" />

Since <InlineMath math="c = ^a\log b" />, then:

<BlockMath math="^m\log b = ^a\log b \cdot ^m\log a" />

Therefore:

<BlockMath math="^a\log b = \frac{^m\log b}{^m\log a}" />

If <InlineMath math="m = b" />, then:

<BlockMath math="^a\log b = \frac{^b\log b}{^b\log a} = \frac{1}{^b\log a}" />

### Chain Rule for Logarithms

**Property 8**: <InlineMath math="^a\log b \times ^b\log c = ^a\log c" />

Proof:
Based on the definition:

<div className="flex flex-col gap-4">
  <BlockMath math="^a\log b = m \Leftrightarrow b = a^m" />
  <BlockMath math="^b\log c = n \Leftrightarrow c = b^n" />
</div>

Substitute the value of <InlineMath math="b" /> into the equation for <InlineMath math="c" />:

<BlockMath math="c = b^n = (a^m)^n = a^{mn}" />

Since <InlineMath math="c = a^{mn}" />, then:

<BlockMath math="^a\log c = ^a\log (a^{mn}) = mn = ^a\log b \times ^b\log c" />

## Application Example

Suppose we want to calculate <InlineMath math="^5\log 125" />.

Using **property 6**:

<BlockMath math="^5\log 125 = ^5\log 5^3 = 3 \cdot ^5\log 5 = 3 \cdot 1 = 3" />

## Exercises

1. Simplify the following expressions:

   a. <InlineMath math="^9\log 81" />

   b. <InlineMath math="^2\log 64 - ^2\log 16" />

   c. <InlineMath math="^4\log 16^{10}" />

2. If <InlineMath math="^5\log 4 = m" />, <InlineMath math="^4\log 3 = n" />, express <InlineMath math="^{12}\log 100" /> in terms of m and n.

3. The population of city A in 2010 was 300,000 people. The average population growth rate is 6% per year. If the population growth is assumed to be the same each year, in how many years will the population of city A become 1 million?

4. How much time is needed for Dini's money, which was initially Rp2,000,000.00, to become Rp6,500,000.00 if she saves it in a bank that gives her an interest rate of 12%?

### Answer Key

1. Determining logarithm values

   a. Answer:

   <div className="flex flex-col gap-4">
     <BlockMath math="^9\log 81 = ^9\log 9^2" />
     <BlockMath math="^9\log 81 = 2 \cdot ^9\log 9" />
     <BlockMath math="^9\log 81 = 2 \cdot 1 = 2" />
   </div>

   b. Answer:

   <div className="flex flex-col gap-4">
     <BlockMath math="^2\log 64 - ^2\log 16 = ^2\log \frac{64}{16}" />
     <BlockMath math="^2\log 64 - ^2\log 16 = ^2\log 4" />
     <BlockMath math="^2\log 64 - ^2\log 16 = 2" />
   </div>

   c. Answer:

   <div className="flex flex-col gap-4">
     <BlockMath math="^4\log 16^{10} = ^4\log (4^2)^{10}" />
     <BlockMath math="^4\log 16^{10} = ^4\log 4^{20}" />
     <BlockMath math="^4\log 16^{10} = 20" />
   </div>

2. Given that <InlineMath math="^5\log 4 = m" />, <InlineMath math="^4\log 3 = n" />

   Then:

   <div className="flex flex-col gap-4">
     <BlockMath math="^{12}\log 100 = \frac{^4\log 100}{^4\log 12}" />
     <BlockMath math="= \frac{^4\log(4 \times 25)}{^4\log(4 \times 3)}" />
     <BlockMath math="= \frac{^4\log 4 + ^4\log 25}{^4\log 4 + ^4\log 3}" />
     <BlockMath math="= \frac{^4\log 4 + 2 \cdot ^4\log 5}{^4\log 4 + ^4\log 3}" />
     <BlockMath math="= \frac{1 + 2 \cdot \frac{1}{m}}{1 + n}" />
     <BlockMath math="= \frac{1 + \frac{2}{m}}{1 + n}" />
   </div>

3. Initial population = 300,000 people

   Annual population growth = 6%

   The appropriate function to describe population growth in <InlineMath math="x" /> years is:

   <BlockMath math="f(x) = 300,000(1 + 0.06)^x" />

   For a population of 1,000,000 people:

   <div className="flex flex-col gap-4">
     <BlockMath math="1,000,000 = 300,000(1 + 0.06)^x" />
     <BlockMath math="1,000,000 = 300,000(1.06)^x" />
     <BlockMath math="\frac{1,000,000}{300,000} = (1.06)^x" />
     <BlockMath math="3.33 = (1.06)^x" />
     <BlockMath math="x = ^{1.06}\log 3.33" />
     <BlockMath math="x = 20.645" />
   </div>

   Therefore, the population will reach 1,000,000 people in 20 or 21 years.

4. Initial savings = Rp2,000,000.00

   Final savings = Rp6,500,000.00

   Interest rate = 12%

   The appropriate function to describe Dini's savings in <InlineMath math="x" /> years is:

   <BlockMath math="f(x) = 2,000,000(1 + 0.12)^x" />

   For a final saving amount of Rp6,500,000.00:

   <div className="flex flex-col gap-4">
     <BlockMath math="6,500,000 = 2,000,000(1 + 0.12)^x" />
     <BlockMath math="6,500,000 = 2,000,000(1.12)^x" />
     <BlockMath math="\frac{6,500,000}{2,000,000} = (1.12)^x" />
     <BlockMath math="3.25 = (1.12)^x" />
     <BlockMath math="x = ^{1.12}\log 3.25" />
     <BlockMath math="x = 10.4" />
   </div>

   Therefore, Dini's savings will reach Rp6,500,000.00 in 10 years.