Solve the Logarithm Product: log₃7 × log₇9 Step-by-Step

Logarithmic Products with Chain Rule Property

log⁡37×log⁡79= \log_37\times\log_79=

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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 multiplication of logarithms
00:12 We will switch between the bases of the logarithms
00:22 We will use this formula in our exercise
00:28 Let's calculate the first logarithm
00:49 This is the solution for the first logarithm, let's substitute in the exercise and solve
01:11 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

log⁡37×log⁡79= \log_37\times\log_79=

2

Step-by-step solution

To solve the expression log⁡37×log⁡79 \log_3 7 \times \log_7 9 , we use a known logarithmic property. This property states that:

log⁡ab×log⁡bc=log⁡ac \log_a b \times \log_b c = \log_a c

Applying this property allows us to simplify:

log⁡37×log⁡79=log⁡39 \log_3 7 \times \log_7 9 = \log_3 9

Next, we need to calculate log⁡39 \log_3 9 . Since 9 can be expressed as 32 3^2 , we have:

log⁡39=log⁡3(32) \log_3 9 = \log_3(3^2)

Using the power rule of logarithms, log⁡b(xn)=n⋅log⁡bx \log_b (x^n) = n \cdot \log_b x , we find:

log⁡3(32)=2⋅log⁡33 \log_3(3^2) = 2 \cdot \log_3 3

Since log⁡33=1 \log_3 3 = 1 , it follows that:

2⋅1=2 2 \cdot 1 = 2

Therefore, the value of log⁡37×log⁡79 \log_3 7 \times \log_7 9 is 2 2 .

The correct answer choice is therefore Choice 3: 2 2 .

3

Final Answer

2 2

Key Points to Remember

Essential concepts to master this topic
  • Chain Rule: log⁡ab×log⁡bc=log⁡ac \log_a b \times \log_b c = \log_a c simplifies products
  • Technique: Transform log⁡37×log⁡79 \log_3 7 \times \log_7 9 into log⁡39 \log_3 9
  • Check: Verify log⁡39=log⁡3(32)=2log⁡33=2 \log_3 9 = \log_3(3^2) = 2 \log_3 3 = 2 ✓

Common Mistakes

Avoid these frequent errors
  • Multiplying logarithms as regular numbers
    Don't calculate log⁡37 \log_3 7 and log⁡79 \log_7 9 separately then multiply = extremely complex decimals! This ignores the chain rule property completely. Always use log⁡ab×log⁡bc=log⁡ac \log_a b \times \log_b c = \log_a c to simplify first.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

What is the chain rule property for logarithms?

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The chain rule states that log⁡ab×log⁡bc=log⁡ac \log_a b \times \log_b c = \log_a c . Notice how the middle base 'b' cancels out, leaving you with a simpler logarithm!

How do I recognize when to use this property?

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Look for multiplication of two logarithms where the argument of the first equals the base of the second. In log⁡37×log⁡79 \log_3 7 \times \log_7 9 , the 7's match up perfectly!

Why does log⁡39 \log_3 9 equal 2?

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Because 9=32 9 = 3^2 ! The logarithm asks: 'What power of 3 gives us 9?' Since 32=9 3^2 = 9 , the answer is 2.

Can I use the change of base formula instead?

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Yes, but it's much more complicated! You'd get ln⁡7ln⁡3×ln⁡9ln⁡7 \frac{\ln 7}{\ln 3} \times \frac{\ln 9}{\ln 7} , which still simplifies to ln⁡9ln⁡3=log⁡39=2 \frac{\ln 9}{\ln 3} = \log_3 9 = 2 . The chain rule is faster!

What if the bases don't line up perfectly?

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Then you cannot use the chain rule directly. You'd need to use other logarithm properties or convert to a common base first. The chain rule only works when there's a perfect 'chain' connection.

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