**This Mathematics Tutorial will focus on LOGARITHMS. We will explain it with examples and give you exercise. At the end of the tutorial, you’ll be able to download it for FREE. Please share this page with your friends who may need it.**

Logarithms are mathematical functions that represent the exponent to which a specific base must be raised to obtain a given number.

They are widely used in various fields, including algebra, calculus, and computer science.

The logarithm of a number *x* to the base *b* is denoted as log* _{b}*(

*x*) or simply log(

*x*) when the base is 10.

Page Contents

### Properties of Logarithms:

**Product Rule**: log(_{b}*xy*) = log(_{b}*x*) + log(_{b}*y*)**Quotient Rule**: log(_{b}*x/y*) = log(_{b}*x*) − log(_{b}*y*)**Power Rule**: log(_{b}*x*)=^{n}*n*⋅ log(_{b})*x***Change of Base Formula**: log(x)=log_{b}(x) / log_{c}(b)_{c}

Note: (⋅) in mathematics means multiply

## Examples:

**Example 1:** Solve for *x* in the equation 2* ^{x}*=8.

**Short solution:** *x *= log_{2}(8)

= log_{2}(2^{3})

= 3

**Long Explanation:**

- Recognize that 2
=8 can be rewritten using logarithms as log^{x}_{2}(8)=*x*. - Apply the definition of a logarithm to write the equation in exponential form: So 2
= 8 is equivalent to^{x}*x*= log_{2}(8). - Evaluate the logarithm: log
_{2}(8) is asking, “*To what power must 2 be raised to get 8*?” - The answer is 3 because 2
^{3}(i.e. 2 x 2 x 2) = 8. - Therefore, the answer is
*x*= 3.

**Example 2:** Simplify the expression log_{5}(25) + log_{5}(⅕).

**Short explanation:**

log_{5}(25) + log_{5}(⅕) = log_{5}(25 ⋅ ⅕)

=log_{5}(5)

=1

HOT TIP: Whenever you see log_{a}(a), for example, log(_{5}5), log(_{2}2), or log(_{7}7) the answer will always be1.

**Long Explanation:**

- Use the product rule of logarithms, which states that log
(_{b}*xy*) = log(_{b}*x*) + log(_{b}*y*). - Apply the
**product rule**to combine the two logarithms: log_{5}(25) + log_{5}(⅕) = log_{5}(25 ⋅ ⅕) - Simplify the expression inside the logarithm: 25 ⋅ ⅕ = 5.
- Therefore, the simplified expression is log
_{5}(5). - Evaluate the logarithm: log
_{5}(25) is asking, “To what power must 5 be raised to get 5?” The answer is**1**. - So, the final result is 1.

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## Exercises:

Make sure you try solving these questions yourself. It’ll help you understand the topic very well.

- Solve for
*x*: 3= 27.^{x} - Simplify: log
_{2}(4) + log_{2}(8) - If log
(_{a}*b*) = 2 and log(_{a}*c*) = 3, find log(_{a}*bc*). - Solve for
*x*:*e*= 20.^{x} - Simplify: 2log
_{3}(5) − log_{3}(125)

## SOLUTIONS TO EXERCISES

Hey, don’t cheat yourself. Make sure you attempt the exercise before checking the solution.

**Exercise 1: **Solve for *x*: 3* ^{x}* = 27.

**Short solution:** *x *= log_{3}(27) = log_{3}(3^{3}) = **3**.

**Long Solution**:

- Apply the definition of logarithms to rewrite the equation as
*x*= log_{3}(27). - Evaluate the logarithm: log
_{3}(27) is asking, “*To what power must 3 be raised to get 27?*” The answer is 3 (because 3 x 3 x 3 is 27). - Therefore, the solution is
*x*= 3.

**Exercise 2: **Simplify: log_{2}(4) + log_{2}(8)

**Short Solution:** log_{2}(4) + log_{2}(8) = log_{2}(4 ⋅ 8) = **log _{2}(32)**

**Long Solution:**

- Use the product rule to combine the two logarithms: log
_{2}(4) + log_{2}(8) = log_{2}(4 ⋅ 8) - Simplify the expression
**inside**the logarithm: 4 ⋅ 8 = 32. - Therefore, the simplified expression is
**log**._{2}(32)

**Exercise 3: **If log* _{a}*(

*b*) = 2 and log

*(*

_{a}*c*) = 3, find log

*(*

_{a}*bc*).

**Short solution**: log* _{a}*(

*bc*) = log

*(b) + log*

_{a}*(*

_{a}*c*) = 2+3 =

**5**

**Long Explanation**:

- Use the product rule of logarithms: log
(_{a}*bc*) = log(b) + log_{a}(_{a}*c*) - Substitute the given values: log
(_{a}*bc*) = 2+3 = 5. - Therefore, log
(_{a}*bc*) =**5**.

**Exercise 4: **Solve for *x*: *e ^{x}* = 20.

**Short solution: ***x *= ln(20) = **2.996**

**Long Explanation:**

*e ^{x}* = 20

Whenever you see this kind of question, just apply the natural logarithm (denoted as

**ln**) to both sides to solve for x

So you will have l**n**(*e*)* ^{x}* =

**ln**(20)

Natural Logarithm always cancels exponential.

So we will just have *x* = **ln**(20)

If you punch **ln**(20) in your calculator, you’ll get **2.996**.

**Exercise 5: **Simplify: 2log_{3}(5) − log_{3}(125)

**Short solution**: 2log_{3}(5) − log_{3}(125) = log_{3} (5^{2} / 5^{3}) = **log _{3} (⅕)**

**Detailed explanation:**

- Use the power rule of logarithms: 2log
_{3}(5) can be written as log_{3}(5^{2}). - Substitute this back into the expression: log
_{3}(5^{2}) − log_{3}(125) - Simplify: 5
^{2}= 25, so the expression becomes log_{3}(25) − log_{3}(125) - Apply the quotient rule: log
_{3}(25) − log_{3}(125) = log_{3}(25 / 125). - Simplify the fraction inside the logarithm: 25 / 125 = 1/5.
- Therefore, the simplified expression is log
_{3}(⅕).

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1 =3, 2=5, 3=5,