Solve: 2log₃4 × log₂9 Logarithm Product Problem

Question

2log34×log29= 2\log_34\times\log_29=

Video Solution

Solution Steps

00:07 Let's solve this problem together.
00:12 We'll start by using the formula for the logarithm of a power. Ready?
00:20 First, let's calculate the power. Here we go!
00:24 Now, we'll apply the formula for multiplying logarithms.
00:30 Next, we'll change the bases of the logarithms. Just follow along.
00:38 We'll use this formula in our exercise. Let's continue.
00:57 Let's calculate the first logarithm now.
01:10 Great! Now, let's do the same for the second logarithm.
01:19 Let's substitute our solutions and finish the calculation.
01:24 And this is how we solve the problem. Well done!

Step-by-Step Solution

To solve this problem, we need to evaluate 2log34×log29 2\log_3 4 \times \log_2 9 . We'll use the change of base formula to simplify the logarithms.

  • Step 1: Apply the change of base formula to both logarithms.
  • Step 2: Simplify the expressions by substituting appropriate values.
  • Step 3: Compute the multiplication of the simplified values.

Step 1: Convert the logarithms using the change of base formula:

log34=log104log103\log_3 4 = \frac{\log_{10} 4}{\log_{10} 3} and log29=log109log102\log_2 9 = \frac{\log_{10} 9}{\log_{10} 2}.

Step 2: Substitute these back into the expression:

2×log104log103×log109log1022 \times \frac{\log_{10} 4}{\log_{10} 3} \times \frac{\log_{10} 9}{\log_{10} 2}.

Recognize that log104=2log102\log_{10} 4 = 2 \log_{10} 2 and log109=2log103\log_{10} 9 = 2 \log_{10} 3, hence simplifying gives:

= 2×2log102log103×2log103log1022 \times \frac{2 \log_{10} 2}{\log_{10} 3} \times \frac{2 \log_{10} 3}{\log_{10} 2}.

Step 3: Cancel terms and calculate:

The terms log102\log_{10} 2 and log103\log_{10} 3 cancel out:

= 2×2×2=82 \times 2 \times 2 = 8.

Therefore, the solution to the problem is 8 \boxed{8} , which corresponds to choice 3 in the provided answer choices.

Answer

8 8