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

Question

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

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!

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

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

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

$\log_38+\log_37^2=$

$\log_38+\log_349=$

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

$\log_3392$