Solve Complex Logarithm Expression: (1/2)log₂4 × log₃8 + log₃9 × log₃7

Q: Why do we calculate log₂4 and log₃9 first?

These are perfect power logarithms! Since $ 2^2 = 4 $ and $ 3^2 = 9 $, both equal 2. This simplifies our expression dramatically before we apply other logarithm rules.

Q: How do I know when to use the product rule for logarithms?

Use $ \log_a(xy) = \log_ax + \log_ay $ when you see addition of logarithms with the same base. In this problem, we get $ \log_38 + \log_349 = \log_3(8 \times 49) $.

Q: What does 2log₃7 become and why?

Using the power rule, $ 2\log_37 = \log_3(7^2) = \log_349 $. The coefficient becomes an exponent inside the logarithm.

Q: How do I multiply 8 × 49 quickly?

Break it down: $ 8 \times 49 = 8 \times (50-1) = 400 - 8 = 392 $. Or use $ 8 \times 7^2 = 8 \times 49 $.

Q: Can I check my answer without a calculator?

Yes! Work backwards: if the answer is $ \log_3392 $, substitute back into the original expression step by step. Each part should simplify to give you confidence in your result.

Logarithm Properties with Product Rules

$\frac{1}{2}\log_24\times\log_38+\log_39\times\log_37=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 Let's solve this problem together.

00:15 First, calculate the logarithm using its definition. Take your time.

00:23 Now, let's isolate the variable X. You're doing great, keep going!

00:27 Next, use the same method to calculate this logarithm. You've got this!

00:36 Here are the solutions for the logarithms. Nice work!

00:42 Now, substitute the values and keep solving. Let's see where it takes us.

00:53 Time to simplify what we can. Break it down step by step.

00:58 Use the power rule for logarithms. Move the two to the exponent. Keep it up!

01:12 Remember the formula for adding logarithms. Go ahead and use it.

01:23 Solve the exponent now. Almost there!

01:31 And that's how we find the solution to our question. Well done!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\frac{1}{2}\log_24\times\log_38+\log_39\times\log_37=$

Step-by-step solution

We break it down into parts

$\log_24=x$

$2^x=4$

$x=2$

$\log_39=x$

$3^x=9$

$x=2$

We substitute into the equation

$\frac{1}{2}\cdot2\log_38+2\log_37=$

$1\cdot\log_38+2\log_37=$

$\log_38+\log_37^2=$

$\log_38+\log_349=$

$\log_3\left(8\cdot49\right)=\log_3392$ $x=2$

Final Answer

$\log_3392$

Key Points to Remember

Essential concepts to master this topic

Rule: Simplify individual logarithms before combining them together
Technique: Use $\log_24 = 2$ since $2^2 = 4$
Check: Verify $\log_3392$ equals original expression by working backwards ✓

Common Mistakes

Avoid these frequent errors

Attempting to combine logarithms before simplifying
Don't try to apply product rules to $\log_24\times\log_38$ directly = messy expressions! This ignores that $\log_24 = 2$ and $\log_39 = 2$ are simple integers. Always evaluate individual logarithms first.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why do we calculate log₂4 and log₃9 first?

These are perfect power logarithms! Since $2^2 = 4$ and $3^2 = 9$ , both equal 2. This simplifies our expression dramatically before we apply other logarithm rules.

How do I know when to use the product rule for logarithms?

Use $\log_a(xy) = \log_ax + \log_ay$ when you see addition of logarithms with the same base. In this problem, we get $\log_38 + \log_349 = \log_3(8 \times 49)$ .

What does 2log₃7 become and why?

Using the power rule, $2\log_37 = \log_3(7^2) = \log_349$ . The coefficient becomes an exponent inside the logarithm.

How do I multiply 8 × 49 quickly?

Break it down: $8 \times 49 = 8 \times (50-1) = 400 - 8 = 392$ . Or use $8 \times 7^2 = 8 \times 49$ .

Can I check my answer without a calculator?

Yes! Work backwards: if the answer is $\log_3392$ , substitute back into the original expression step by step. Each part should simplify to give you confidence in your result.

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Solve Complex Logarithm Expression: (1/2)log₂4 × log₃8 + log₃9 × log₃7

Logarithm Properties with Product Rules

❤️ 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 calculate log₂4 and log₃9 first?

How do I know when to use the product rule for logarithms?

What does 2log₃7 become and why?

How do I multiply 8 × 49 quickly?

Can I check my answer without a calculator?

🌟 Unlock Your Math Potential

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