# Nakafa Framework: LLM

URL: /en/subject/high-school/11/mathematics/function-modeling/logarithmic-function-identity
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---

export const metadata = {
  title: "Logarithmic Function Identity",
  description: "Master logarithmic identities including product, quotient, power, and change of base formulas. Solve equations with practical real-world examples.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/18/2025",
  subject: "Functions and Their Modeling",
};

## Understanding Logarithmic Identities

Logarithmic identities are special properties that apply to all logarithmic functions. These properties are very helpful in simplifying calculations and solving complex logarithmic equations.

Before discussing logarithmic identities, let's recall that logarithms are the inverse of exponents. If <InlineMath math="b^x = a" />, then <InlineMath math="^b\log a = x" />.

## Basic Logarithmic Identities

### Product Identity

<BlockMath math="^b\log(MN) = {^b\log M} + {^b\log N}" />

The logarithm of a product equals the sum of the logarithms of each number.

**Example:**

<BlockMath math="^2\log(8 \times 4) = {^2\log 8} + {^2\log 4} = 3 + 2 = 5" />

### Quotient Identity

<BlockMath math="^b\log\left(\frac{M}{N}\right) = {^b\log M} - {^b\log N}" />

The logarithm of a quotient equals the difference between the logarithm of the numerator and the logarithm of the denominator.

**Example:**

<BlockMath math="^3\log\left(\frac{81}{9}\right) = {^3\log 81} - {^3\log 9} = 4 - 2 = 2" />

### Power Identity

<BlockMath math="^b\log(M^p) = p \cdot {^b\log M}" />

The logarithm of a number raised to a power equals the power multiplied by the logarithm of that number.

**Example:**

<BlockMath math="^2\log(4^3) = 3 \cdot {^2\log 4} = 3 \times 2 = 6" />

## Special Logarithmic Identities

### Change of Base

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

This identity allows us to change the logarithm base as needed.

**Example:**

<BlockMath math="^3\log 9 = \frac{^{10}\log 9}{^{10}\log 3} = \frac{0.954}{0.477} = 2" />

### Equality Identity

If <InlineMath math="^b\log M = {^b\log N}" />, then <InlineMath math="M = N" />

Two numbers that have the same logarithmic value (with the same base) must be the same number.

### Inequality Identity

- If <InlineMath math="b > 1" /> and <InlineMath math="^b\log M < {^b\log N}" />, then <InlineMath math="M < N" />
- If <InlineMath math="0 < b < 1" /> and <InlineMath math="^b\log M < {^b\log N}" />, then <InlineMath math="M > N" />

## Applications of Logarithmic Identities

### Simplifying Expressions

Simplify <InlineMath math="^2\log 32 + {^2\log 8} - {^2\log 4}" />

**Solution:**

<div className="flex flex-col gap-4">
  <BlockMath math="^2\log 32 + {^2\log 8} - {^2\log 4}" />
  <BlockMath math="= {^2\log(32 \times 8)} - {^2\log 4}" />
  <BlockMath math="= {^2\log 256} - {^2\log 4}" />
  <BlockMath math="= {^2\log\left(\frac{256}{4}\right)}" />
  <BlockMath math="= {^2\log 64} = 6" />
</div>

### Solving Equations

Find the value of <InlineMath math="x" /> if <InlineMath math="^3\log x + {^3\log 9} = {^3\log 81}" />

**Solution:**

<div className="flex flex-col gap-4">
  <BlockMath math="^3\log x + {^3\log 9} = {^3\log 81}" />
  <BlockMath math="^3\log(x \times 9) = {^3\log 81}" />
  <BlockMath math="x \times 9 = 81" />
  <BlockMath math="x = \frac{81}{9} = 9" />
</div>

## Real-Life Applications

### Richter Scale

Earthquake strength is measured using the Richter scale which is based on logarithms:

<BlockMath math="R = \log\left(\frac{I}{I_0}\right)" />

Where:

- <InlineMath math="R" /> = Richter scale value
- <InlineMath math="I" /> = earthquake intensity
- <InlineMath math="I_0" /> = reference intensity (zero level)

**Example:** An earthquake that occurred in Haiti in 2010 had an intensity of <InlineMath math="10^7" /> times compared to zero-level earthquakes. What is the Richter scale strength of that earthquake?

**Solution:**

<div className="flex flex-col gap-4">
  <BlockMath math="R = \log\left(\frac{I}{I_0}\right)" />
  <BlockMath math="= \log\left(\frac{10^7 I_0}{I_0}\right)" />
  <BlockMath math="= \log(10^7)" />
  <BlockMath math="= 7" />
</div>

Therefore, the earthquake in Haiti in 2010 had a strength of 7 on the Richter scale.

### Battery Charging

Battery charging time can be calculated using the logarithmic formula:

<BlockMath math="t = -\frac{1}{k}\ln\left(1 - \frac{C}{C_0}\right)" />

Where:

- <InlineMath math="t" /> = charging time (in minutes)
- <InlineMath math="k" /> = charging constant
- <InlineMath math="C" /> = desired capacity
- <InlineMath math="C_0" /> = maximum capacity

**Example:** Determine the time required to charge a battery from empty to 90% full. Assume <InlineMath math="k = 0.02" />.

**Solution:**

<div className="flex flex-col gap-4">
  <BlockMath math="t = -\frac{1}{k}\ln\left(1 - \frac{C}{C_0}\right)" />
  <BlockMath math="= -\frac{1}{0.02}\ln\left(1 - \frac{0.9C_0}{C_0}\right)" />
  <BlockMath math="= -50\ln(1 - 0.9)" />
  <BlockMath math="= -50\ln(0.1)" />
  <BlockMath math="\approx 115.13" />
</div>

Therefore, the charging time is approximately 115 minutes.

### Car Price Depreciation

Logarithmic functions are also used for modeling decay/depreciation with the formula:

<BlockMath math="H(t) = ce^{kt}" />

where <InlineMath math="H(t)" /> is the value at time <InlineMath math="t" />.

**Example:** At any given time, the price of a used car is not proportional to its current price. If a new car costs 200 million rupiah and after 5 years becomes 100 million rupiah, determine the car's price after 10 years of use.

**Solution:**

<div className="flex flex-col gap-4">
  <BlockMath math="H(0) = 200 \text{ million, so } 200 = ce^0 = c" />
  <BlockMath math="H(5) = 100 \text{ million, so } 100 = 200e^{5k}" />
  <BlockMath math="e^{5k} = \frac{1}{2} \text{, therefore } 5k = \ln\left(\frac{1}{2}\right)" />
  <BlockMath math="k = \frac{1}{5}\ln\left(\frac{1}{2}\right) = -0.1386" />
</div>

From these results, the car's price at any time <InlineMath math="t" /> is:

<BlockMath math="H(t) = 200e^{-0.1386t}" />

Therefore, the car's price after 10 years of use is:

<BlockMath math="H(10) = 200e^{-0.1386(10)} = 200e^{-1.386} \approx 50 \text{ million rupiah}" />

## Exercises

**Problem 1**

Simplify: <InlineMath math="^5\log 125 + {^5\log 25} - {^5\log 5}" />

**Problem 2**

If <InlineMath math="^2\log x = 3" /> and <InlineMath math="^2\log y = 5" />, find the value of <InlineMath math="^2\log(xy)" />

**Problem 3**

Find the value of <InlineMath math="x" /> if <InlineMath math="^4\log(x-1) = {^4\log 16} - {^4\log 2}" />

### Answer Key

**Answer 1**

<div className="flex flex-col gap-4">
  <BlockMath math="^5\log 125 + {^5\log 25} - {^5\log 5}" />
  <BlockMath math="= 3 + 2 - 1 = 4" />
</div>

**Answer 2**

<div className="flex flex-col gap-4">
  <BlockMath math="^2\log(xy) = {^2\log x} + {^2\log y}" />
  <BlockMath math="= 3 + 5 = 8" />
</div>

**Answer 3**

<div className="flex flex-col gap-4">
  <BlockMath math="^4\log(x-1) = {^4\log 16} - {^4\log 2}" />
  <BlockMath math="^4\log(x-1) = {^4\log\left(\frac{16}{2}\right)}" />
  <BlockMath math="^4\log(x-1) = {^4\log 8}" />
  <BlockMath math="x - 1 = 8" />
  <BlockMath math="x = 9" />
</div>