Back to: MATHEMATICS SS3

**Welcome to class! **

In today’s class, we will be talking about the theory of logarithms. Enjoy the class!

**Logarithm Theory**

**Rules of logarithms**

**1) Product Rule:**

The **logarithm of a product** is the **sum** of the logarithms of the factors.

log_{a} *xy* = log_{a} *x* + log_{a} y

**2) Quotient Rule:**

The** logarithm of a quotient** is the logarithm of the numerator **minus** the logarithm of the denominator

log* _{a }(x/y)* = log

*– log*

_{a}x_{a}

*y*

**3) Power Rule:**

log* _{a} x^{n} *=

*n*log

_{a}x**4) Change of Base Rule:**

*log*_{a }*b = log*_{c }*b / log _{c}*

*a*

*log*_{a }*b = 1 / log _{b}*

*a*

where* x *and *y* are positive, and *a* > 0, *a* ≠ 1

**Example:**

Express log _{2} 4 + log _{2} 5 as a single logarithm

*Solution*

log _{2} 4 + log _{2} 5 = log _{2} (4 × 5) = log _{2} 20

**The properties of logarithms**

1. log_{a} 1 = 0 since a^{0} = 1.

2. log_{a} a = 1 since a^{1} = a.

3. log_{a} a^{x} = x since a^{x} = a^{x}.

**Product property:** The log of a product equals the sum of the logs.

**Quotient Property:** The log of a quotient equals the difference of the logs.

**Power Property:** The log of power equals the product of the power and the log.

**Logarithms applications**

- Population growth
- Interest Rate Problems
- Mortgage Problems
- Radioactive Decay Problems
- Earthquake Problems

**Indices**

**Laws of Indices:**

**Fractional indices:**

In our next class, we will be talking about **Surds**. We hope you enjoyed the class.

Should you have any further question, feel free to ask in the comment section below and trust us to respond as soon as possible.

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