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

12log⁡24×log⁡38+log⁡39×log⁡37= \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
1

Understand the problem

12log⁡24×log⁡38+log⁡39×log⁡37= \frac{1}{2}\log_24\times\log_38+\log_39\times\log_37=

2

Step-by-step solution

We break it down into parts

log⁡24=x \log_24=x

2x=4 2^x=4

x=2 x=2

log⁡39=x \log_39=x

3x=9 3^x=9

x=2 x=2

We substitute into the equation

12⋅2log⁡38+2log⁡37= \frac{1}{2}\cdot2\log_38+2\log_37=

1⋅log⁡38+2log⁡37= 1\cdot\log_38+2\log_37=

log⁡38+log⁡372= \log_38+\log_37^2=

log⁡38+log⁡349= \log_38+\log_349=

log⁡3(8⋅49)=log⁡3392 \log_3\left(8\cdot49\right)=\log_3392 x=2 x=2

3

Final Answer

log⁡3392 \log_3392

Practice Quiz

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\( \log_{10}3+\log_{10}4= \)

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