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Learn eight logarithm properties: product, quotient, power, and change of base. Use them to solve logarithmic equations.

---

## 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 $$a > 0$$ and $$a \neq 1$$, $$b > 0$$, $$c > 0$$, $$m > 0$$, $$m \neq 1$$, where $$a, b, c, m, n$$ are real numbers $$(a, b, c, m, n \in \mathbb{R})$$. The following are **logarithm properties**:

Visible text: Let and , , , , , where are real numbers . The following are **logarithm properties**:

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

Visible text: 1. 
2. 
3. 
4. 
5. 
6. 
7. 
8.

## Proving Logarithm Properties

### Logarithm of Multiplication

**Property** $$4$$: $$^a\log (b \times c) = ^a\log b + ^a\log c$$

Visible text: **Property** :

Proof: Let $$^a\log b = m$$ and $$^a\log c = n$$

Visible text: Proof: Let and

This means:

$$b = a^m$$ and $$c = a^n$$

Visible text: and

Using the property of exponents:

```math
b \times c = a^m \times a^n = a^{m+n}
```

Therefore:

```math
^a\log (b \times c) = ^a\log (a^{m+n}) = m + n = ^a\log b + ^a\log c
```

### Logarithm of Division

**Property** $$5$$: $$^a\log \left(\frac{b}{c}\right) = ^a\log b - ^a\log c$$

Visible text: **Property** :

Proof: Let $$^a\log b = m$$ and $$^a\log c = n$$

Visible text: Proof: Let and

Then $$b = a^m$$ and $$c = a^n$$

Visible text: Then and

Recall that $$\frac{a^m}{a^n} = a^{m-n}$$, so:

Visible text: Recall that , so:

```math
\frac{b}{c} = \frac{a^m}{a^n} = a^{m-n}
```

Therefore:

```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** $$6$$: $$^a\log b^n = n \cdot ^a\log b$$

Visible text: **Property** :

Proof: Let $$^a\log b = m$$

Visible text: Proof: Let

$$^a\log b^n$$ means the logarithm of $$b$$ raised
to the power of $$n$$

Visible text: means the logarithm of raised
to the power of

$$^a\log b^n = ^a\log (\underbrace{b \times b \times b \times \ldots \times b}_{n \text{ factors}})$$

Using property $$4$$ repeatedly:

Visible text: Using property repeatedly:

```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** $$7$$: $$^a\log b = \frac{^m\log b}{^m\log a} = \frac{1}{^b\log a}$$

Visible text: **Property** :

Proof:
Based on the definition of logarithm, $$^a\log b = c$$ if and only if $$b = a^c$$

Visible text: Proof:
Based on the definition of logarithm, if and only if

Suppose we use base $$m$$ for the logarithm of $$b$$:

Visible text: Suppose we use base for the logarithm of :

```math
^m\log b = ^m\log a^c
```

Using property $$6$$:

Visible text: Using property :

```math
^m\log b = c \cdot ^m\log a
```

Since $$c = ^a\log b$$, then:

Visible text: Since , then:

```math
^m\log b = ^a\log b \cdot ^m\log a
```

Therefore:

```math
^a\log b = \frac{^m\log b}{^m\log a}
```

If $$m = b$$, then:

Visible text: If , then:

```math
^a\log b = \frac{^b\log b}{^b\log a} = \frac{1}{^b\log a}
```

### Chain Rule for Logarithms

**Property** $$8$$: $$^a\log b \times ^b\log c = ^a\log c$$

Visible text: **Property** :

Proof:
Based on the definition:

Component: MathContainer
Children:

```math
^a\log b = m \Leftrightarrow b = a^m
```

```math
^b\log c = n \Leftrightarrow c = b^n
```

Substitute the value of $$b$$ into the equation for $$c$$:

Visible text: Substitute the value of into the equation for :

```math
c = b^n = (a^m)^n = a^{mn}
```

Since $$c = a^{mn}$$, then:

Visible text: Since , then:

```math
^a\log c = ^a\log (a^{mn}) = mn = ^a\log b \times ^b\log c
```

## Application Example

Suppose we want to calculate $$^5\log 125$$.

Visible text: Suppose we want to calculate .

Using **property** $$6$$:

Visible text: Using **property** :

```math
^5\log 125 = ^5\log 5^3 = 3 \cdot ^5\log 5 = 3 \cdot 1 = 3
```

## Exercises

1. Simplify the following expressions:

    1. $$^9\log 81$$

    2. $$^2\log 64 - ^2\log 16$$

    3. $$^4\log 16^{10}$$

2. If $$^5\log 4 = m$$, $$^4\log 3 = n$$, express $$^{12}\log 100$$ in terms of $$m$$ and $$n$$.

3. The population of city $$A$$ in $$2010$$ was $$300{,}000 \text{ people}$$. 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 $$1 \text{ million}$$?

4. How much time is needed for Dini's money, which was initially $$\text{Rp}2{,}000{,}000.00$$, to become $$\text{Rp}6{,}500{,}000.00$$ if she saves it in a bank that gives her an interest rate of $$12\%$$?

Visible text: 1. Simplify the following expressions:

 1. 

 2. 

 3. 

2. If , , express in terms of and .

3. The population of city in was . The average population growth rate is per year. If the population growth is assumed to be the same each year, in how many years will the population of city become ?

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

### Answer Key

1. Determining logarithm values

    1. Answer:

        <MathContainer>
          
       
       ```math
       ^9\log 81 = ^9\log 9^2
       ```

          
       
       ```math
       ^9\log 81 = 2 \cdot ^9\log 9
       ```

          
       
       ```math
       ^9\log 81 = 2 \cdot 1 = 2
       ```

        </MathContainer>

    2. Answer:

        <MathContainer>
          
       
       ```math
       ^2\log 64 - ^2\log 16 = ^2\log \frac{64}{16}
       ```

          
       
       ```math
       ^2\log 64 - ^2\log 16 = ^2\log 4
       ```

          
       
       ```math
       ^2\log 64 - ^2\log 16 = 2
       ```

        </MathContainer>

    3. Answer:

        <MathContainer>
          
       
       ```math
       ^4\log 16^{10} = ^4\log (4^2)^{10}
       ```

          
       
       ```math
       ^4\log 16^{10} = ^4\log 4^{20}
       ```

          
       
       ```math
       ^4\log 16^{10} = 20
       ```

        </MathContainer>

2. Given that $$^5\log 4 = m$$, $$^4\log 3 = n$$

   Then:

   <MathContainer>
     
   
   ```math
   ^{12}\log 100 = \frac{^4\log 100}{^4\log 12}
   ```

     
   
   ```math
   = \frac{^4\log(4 \times 25)}{^4\log(4 \times 3)}
   ```

     
   
   ```math
   = \frac{^4\log 4 + ^4\log 25}{^4\log 4 + ^4\log 3}
   ```

     
   
   ```math
   = \frac{^4\log 4 + 2 \cdot ^4\log 5}{^4\log 4 + ^4\log 3}
   ```

     
   
   ```math
   = \frac{1 + 2 \cdot \frac{1}{m}}{1 + n}
   ```

     
   
   ```math
   = \frac{1 + \frac{2}{m}}{1 + n}
   ```

   </MathContainer>

3. The initial population is $$300{,}000 \text{ people}$$

   The annual population growth is $$6\%$$.

   The appropriate function to describe population growth in $$x \text{ years}$$ is:

   
   
   ```math
   f(x) = 300{,}000(1 + 0.06)^x
   ```

   For a population of $$1{,}000{,}000 \text{ people}$$:

   <MathContainer>
     
   
   ```math
   1{,}000{,}000 = 300{,}000(1 + 0.06)^x
   ```

     
   
   ```math
   1{,}000{,}000 = 300{,}000(1.06)^x
   ```

     
   
   ```math
   \frac{1{,}000{,}000}{300{,}000} = (1.06)^x
   ```

     
   
   ```math
   3.33 = (1.06)^x
   ```

     
   
   ```math
   x = ^{1.06}\log 3.33
   ```

     
   
   ```math
   x = 20.645
   ```

   </MathContainer>

   Therefore, the population will reach $$1{,}000{,}000 \text{ people}$$ in $$20$$ or $$21 \text{ years}$$.

4. The initial savings are $$\text{Rp}2{,}000{,}000.00$$

   The final savings are $$\text{Rp}6{,}500{,}000.00$$

   The interest rate is $$12\%$$.

   The appropriate function to describe Dini's savings in $$x \text{ years}$$ is:

   
   
   ```math
   f(x) = 2{,}000{,}000(1 + 0.12)^x
   ```

   For a final saving amount of $$\text{Rp}6{,}500{,}000.00$$:

   <MathContainer>
     
   
   ```math
   6{,}500{,}000 = 2{,}000{,}000(1 + 0.12)^x
   ```

     
   
   ```math
   6{,}500{,}000 = 2{,}000{,}000(1.12)^x
   ```

     
   
   ```math
   \frac{6{,}500{,}000}{2{,}000{,}000} = (1.12)^x
   ```

     
   
   ```math
   3.25 = (1.12)^x
   ```

     
   
   ```math
   x = ^{1.12}\log 3.25
   ```

     
   
   ```math
   x = 10.4
   ```

   </MathContainer>

   Therefore, Dini's savings will reach $$\text{Rp}6{,}500{,}000.00$$ in $$10 \text{ years}$$.

Visible text: 1. Determining logarithm values

 1. Answer:

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 </MathContainer>

 2. Answer:

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 </MathContainer>

 3. Answer:

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 </MathContainer>

2. Given that , 

 Then:

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 </MathContainer>

3. The initial population is 

 The annual population growth is .

 The appropriate function to describe population growth in is:

 
 

 For a population of :

 <MathContainer>
 
 

 
 

 
 

 
 

 
 

 
 

 </MathContainer>

 Therefore, the population will reach in or .

4. The initial savings are 

 The final savings are 

 The interest rate is .

 The appropriate function to describe Dini's savings in is:

 
 

 For a final saving amount of :

 <MathContainer>
 
 

 
 

 
 

 
 

 
 

 
 

 </MathContainer>

 Therefore, Dini's savings will reach in .