Free Maths Tutorial: Logarithms [Explained with Examples]

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.

Properties of Logarithms:

1. Product Rule: log_b​​(xy) = log_b​​(x) + log_b​​(y)
2. Quotient Rule: log_b​​(x/y​) = log_b​​(x) − log_b​​(y)
3. Power Rule: log_b​​(xⁿ)=n ⋅ log_b​​(x)
4. Change of Base Formula: log_b​​(x)=log_c​(x) / log_c​(b)​

Note: (⋅) in mathematics means multiply

Examples:

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

Short solution:
x = log₂​(8)
= log₂​(2³)
= 3

Long Explanation:

1. Recognize that 2^x=8 can be rewritten using logarithms as log⁡₂(8)=x.
2. Apply the definition of a logarithm to write the equation in exponential form: So 2^x = 8 is equivalent to x = log₂​(8).
3. Evaluate the logarithm: log_2​(8) is asking, “To what power must 2 be raised to get 8?”
4. The answer is 3 because 2³ (i.e. 2 x 2 x 2) = 8.
5. 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)
=log₅​(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 be 1.

Long Explanation:

1. Use the product rule of logarithms, which states that log_b​(xy) = log_b​(x) + log_b​(y).
2. Apply the product rule to combine the two logarithms: log⁡₅(25) + log⁡₅(⅕) = log⁡₅(25 ⋅ ⅕)
3. Simplify the expression inside the logarithm: 25 ⋅ ⅕ = 5.
4. Therefore, the simplified expression is log⁡₅(5).
5. Evaluate the logarithm: log⁡₅(25) is asking, “To what power must 5 be raised to get 5?” The answer is 1.
6. 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.

1. Solve for x: 3^x = 27.
2. Simplify: log⁡₂(4) + log⁡₂(8)
3. If log_a​(b) = 2 and log_a​(c) = 3, find log_a​(bc).
4. Solve for x: e^x = 20.
5. Simplify: 2log⁡₃(5) − log⁡₃(125)

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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​(3³) = 3.

Long Solution:

1. Apply the definition of logarithms to rewrite the equation as x = log₃​(27).
2. 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).
3. 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:

1. Use the product rule to combine the two logarithms: log⁡₂(4) + log⁡₂(8) = log⁡₂(4 ⋅ 8)
2. Simplify the expression inside the logarithm: 4 ⋅ 8 = 32.
3. 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_a​(b) + log_a​(c) = 2+3 = 5

Long Explanation:

1. Use the product rule of logarithms: log_a​(bc) = log_a​(b) + log_a​(c)
2. Substitute the given values: log_a​(bc) = 2+3 = 5.
3. 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 ln⁡(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 (5² / 5³) = log⁡₃ (⅕)

Detailed explanation:

1. Use the power rule of logarithms: 2log⁡₃(5) can be written as log₃​(5²).
2. Substitute this back into the expression: log⁡₃(5²) − log₃(125)
3. Simplify: 5² = 25, so the expression becomes log₃(25) − log₃(125)
4. Apply the quotient rule: log₃(25) − log₃(125) = log₃​(25 / 125​).
5. Simplify the fraction inside the logarithm: 25 / 125 = 1/5​.
6. Therefore, the simplified expression is log₃(⅕).

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