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

Question

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

00:00 Solve

00:05 Calculate the logarithm according to its definition

00:13 Isolate and find X

00:17 Use the same method and calculate this logarithm

00:26 These are the solutions for the logarithms

00:32 Substitute and continue solving

00:43 Simplify what we can

00:48 Use the power rule for logarithms and move the 2 to the exponent

01:02 Use the formula for adding logarithms

01:13 Solve the exponent

01:21 And this is the solution to the question

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

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

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

$\log_38+\log_37^2=$

$\log_38+\log_349=$

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

$\log_3392$