Solve log₂4 + log₂5: Adding Logarithms with Base 2

Q: What if the bases were different?

If bases are different (like $ \log_2 4 + \log_3 5 $), you cannot combine them using the product rule. The bases must be identical to use this property.

Q: How do I remember when to multiply vs add?

Memory trick: Adding logs = multiplying arguments, while multiplying logs = raising to powers. The operations 'flip' in a sense!

Q: Can I simplify log₂20 further?

You could factor it: $ \log_2 20 = \log_2 (4 \times 5) = \log_2 4 + \log_2 5 $ (which brings us back to the original!). Or since $ 20 = 4 \times 5 = 2^2 \times 5 $, we get $ \log_2 20 = 2 + \log_2 5 $.

Q: Why is this rule useful?

The product rule helps simplify complex expressions and solve logarithmic equations. It's also the reverse of the power rule, making it essential for logarithmic calculations!

Logarithm Addition with Product Rule

$\log_24+\log_25=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:04 We will use the formula for adding logarithms

00:11 We will use this formula in our exercise

00:21 We can see that we can use it, the bases are equal

00:31 Let's solve the parentheses

00:39 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\log_24+\log_25=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the given expression as $\log_2 4 + \log_2 5$ .
Step 2: Use the sum of logarithms rule to simplify the expression.
Step 3: Calculate the product and express the result.

Let's work through each step:

Step 1: We have $\log_2 4 + \log_2 5$ as our expression.

Step 2: Apply the sum of logarithms formula:

$\log_2 4 + \log_2 5 = \log_2 (4 \cdot 5)$

Step 3: Calculate the product:

$4 \times 5 = 20$

Thus, $\log_2 (4 \cdot 5) = \log_2 20$ .

Therefore, the solution to the problem is $\log_2 20$ .

Final Answer

$\log_220$

Key Points to Remember

Essential concepts to master this topic

Product Rule: $\log_b x + \log_b y = \log_b (xy)$ for same bases
Technique: Calculate $\log_2 4 + \log_2 5 = \log_2 (4 \times 5) = \log_2 20$
Check: Verify bases match before applying rule: both are base 2 ✓

Common Mistakes

Avoid these frequent errors

Adding the arguments directly instead of multiplying
Don't add 4 + 5 = 9 to get $\log_2 9$ ! This ignores the logarithm product rule and gives a completely wrong answer. Always multiply the arguments when adding logarithms with the same base: $\log_2 4 + \log_2 5 = \log_2 (4 \times 5) = \log_2 20$ .

Practice Quiz

Test your knowledge with interactive questions

$\log_{10}3+\log_{10}4=$

FAQ

Everything you need to know about this question

Why can't I just add 4 + 5 to get 9?

Because logarithms follow special rules! When you add logarithms with the same base, you multiply their arguments, not add them. Think of it as: $\log_2 4 + \log_2 5 = \log_2 (4 \times 5) = \log_2 20$ .

What if the bases were different?

If bases are different (like $\log_2 4 + \log_3 5$ ), you cannot combine them using the product rule. The bases must be identical to use this property.

How do I remember when to multiply vs add?

Memory trick: Adding logs = multiplying arguments, while multiplying logs = raising to powers. The operations 'flip' in a sense!

Can I simplify log₂20 further?

You could factor it: $\log_2 20 = \log_2 (4 \times 5) = \log_2 4 + \log_2 5$ (which brings us back to the original!). Or since $20 = 4 \times 5 = 2^2 \times 5$ , we get $\log_2 20 = 2 + \log_2 5$ .

Why is this rule useful?

The product rule helps simplify complex expressions and solve logarithmic equations. It's also the reverse of the power rule, making it essential for logarithmic calculations!

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Solve log₂4 + log₂5: Adding Logarithms with Base 2

Logarithm Addition with Product Rule

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't I just add 4 + 5 to get 9?

What if the bases were different?

How do I remember when to multiply vs add?

Can I simplify log₂20 further?

Why is this rule useful?

🌟 Unlock Your Math Potential

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