Solve log₁₀3 + log₁₀4: Step-by-Step Logarithm Addition

Q: Why do we multiply 3 and 4 instead of adding them?

The logarithm addition property says $ \log_b(x) + \log_b(y) = \log_b(x \cdot y) $. This means when you add logarithms, you multiply their arguments!

Q: What if the logarithms have different bases?

You cannot use this property! Both logarithms must have the same base. If bases differ, you need to convert them first using change of base formula.

Q: Can I use a calculator to check my answer?

Yes! Calculate $ \log_{10}3 + \log_{10}4 $ separately, then calculate $ \log_{10}12 $. Both should give you approximately 1.079.

Q: Does this work for subtraction too?

Yes, but with division! $ \log_b(x) - \log_b(y) = \log_b(\frac{x}{y}) $. So subtracting logarithms means dividing their arguments.

Q: Why is the answer still a logarithm?

The result $ \log_{10}12 $ is the exact form. You could approximate it as 1.079, but keeping it as $ \log_{10}12 $ is more precise and often preferred in mathematics.

Logarithm Addition with Base-10 Properties

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 We'll use the formula for adding logarithms

00:16 We'll use this formula in our exercise

00:23 We can see it's applicable, bases are equal

00:39 Let's solve the parentheses

00:44 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

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

Step-by-step solution

To solve this problem, we will use the property of logarithms that allows us to combine the sum of two logarithms:

Step 1: Identify the formula. We use the property $\log_b(x) + \log_b(y) = \log_b(x \cdot y)$ where both logarithms must have the same base.
Step 2: Recognize the base. Here, both logarithms are in base 10: $\log_{10}3$ and $\log_{10}4$ .
Step 3: Apply the property. Add the two logarithms using the formula: $\log_{10}3 + \log_{10}4 = \log_{10}(3 \cdot 4)$ .
Step 4: Perform the multiplication. Compute $3 \cdot 4$ to get 12.
Step 5: Express the result as a single logarithm: $\log_{10}12$ .

Therefore, the expression $\log_{10}3 + \log_{10}4$ simplifies to $\log_{10}12$ .

Final Answer

$\log_{10}12$

Key Points to Remember

Essential concepts to master this topic

Addition Property: $\log_b(x) + \log_b(y) = \log_b(x \cdot y)$ when bases match
Technique: Convert $\log_{10}3 + \log_{10}4$ to $\log_{10}(3 \times 4) = \log_{10}12$
Verification: Check that both original logarithms have same base before applying property ✓

Common Mistakes

Avoid these frequent errors

Adding the numbers inside the logarithms instead of multiplying
Don't calculate $\log_{10}3 + \log_{10}4 = \log_{10}7$ ! This treats logarithms like regular addition, giving completely wrong results. Always multiply the arguments when adding logarithms: $\log_{10}(3 \times 4) = \log_{10}12$ .

Practice Quiz

Test your knowledge with interactive questions

$\log_75-\log_72=$

FAQ

Everything you need to know about this question

Why do we multiply 3 and 4 instead of adding them?

The logarithm addition property says $\log_b(x) + \log_b(y) = \log_b(x \cdot y)$ . This means when you add logarithms, you multiply their arguments!

What if the logarithms have different bases?

You cannot use this property! Both logarithms must have the same base. If bases differ, you need to convert them first using change of base formula.

Can I use a calculator to check my answer?

Yes! Calculate $\log_{10}3 + \log_{10}4$ separately, then calculate $\log_{10}12$ . Both should give you approximately 1.079.

Does this work for subtraction too?

Yes, but with division! $\log_b(x) - \log_b(y) = \log_b(\frac{x}{y})$ . So subtracting logarithms means dividing their arguments.

Why is the answer still a logarithm?

The result $\log_{10}12$ is the exact form. You could approximate it as 1.079, but keeping it as $\log_{10}12$ is more precise and often preferred in mathematics.

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Solve log₁₀3 + log₁₀4: Step-by-Step Logarithm Addition

Logarithm Addition with Base-10 Properties

❤️ 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 do we multiply 3 and 4 instead of adding them?

What if the logarithms have different bases?

Can I use a calculator to check my answer?

Does this work for subtraction too?

Why is the answer still a logarithm?

🌟 Unlock Your Math Potential

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