A non-standard logarithmic equation

This video is about an extraordinary logarithmic equation whose solution method is unique.
challenging math problems, logarithmic equations, substitution, non-standard problem.

[robertlee]

When you solve the equation log_2(x)=3-x, we can turn that into an exponential equation as 2^(3-x)=x. Using properties of exponents, that equation turns into 8=x2^x. We can change base 2 as x(e^ln(2))^x=8 or xe^(xln(2))=8. We can use the Lambert W function. Let's multiply both sides by ln(2). That makes this as (ln(2))xe^(xln(2))=8ln(2). Let's do substitution by letting u=xln(2), which ue^u=8ln(2). Take W on both sides and substitute back will give xln(2)=2ln(2). Therefore, x=2.

[justabunga]

In the equation where you solved the quadratic formula with the negative square root sign, we can solve the equation without using the graph or by trial and error. This is called the Lambert W function. log_2(x)=3-x can be turned into an exponential equation as 2^(3-x)=x. Using properties of exponents and cross multiply this as x2^x=8. Turn base 2 into base e as xe^(xln(2))=8. Multiply both sides by ln(2) as xln(2)e^(xln(2))=8ln(2). Take W on both sides, which turns the equation as xln(2)=W(8ln(2)), so x=W(8ln(2))/ln(2)=2.

[phaseshift]

Is there another method to get the other value of x without using graph. ?!

[yqsixzi]

What does it mean infinity over zero in calculus ?
Answer is zero or infinity?