Solve each system. Use any method you wish. { ln x=5 ln y log2 x=3+2 log2 y .

Name: Problem 54, Chapter 8: College Algebra
Uploaded: 2020-06-25T19:48:18-08:00
Duration: 7 min 13 s
Channel: Sushmit Acharya
Description: Solve each system. Use any method you wish. { ln x=5 ln y log2 x=3+2 log2 y .

Solve each system. Use any method you wish. { ln x=5 ln...

Key Concepts

Systems of Equations

Systems of equations involve solving multiple equations simultaneously to find common solutions. Techniques such as substitution or elimination are applied, and when the system contains logarithmic expressions, one can use logarithmic properties to reframe the equations into a more solvable form.

Exponential Equations

Exponential equations are equations in which variables appear as exponents. Converting logarithmic equations into exponential form leverages the one-to-one nature of the exponential and logarithmic functions, facilitating the solution of the equations by isolating the variable.

Properties of Logarithms

The properties of logarithms, such as the power rule, product rule, and change of base formula, are fundamental in manipulating and simplifying logarithmic expressions. These rules allow for the transformation of equations, for instance by moving coefficients to exponents or combining multiple logarithmic terms into a single term.

Logarithmic Functions

Logarithmic functions serve as inverses of exponential functions, providing a key tool for transforming multiplicative relationships into additive ones, which simplifies the process of solving equations that involve exponents and power laws.

Transcript

00:01 All right, here we have a system of nonlinear equations with logarithms.

00:07 As you can tell from here, the best thing to do for these situations is just try to get rid of the logarithms by using just simple logarithmic rules and then just solve for them after getting an equation with just x and y in them.

00:24 So let's just start with transforming the top equation into a, let's say, normal equation.

00:29 So we have ln of x equals 5 allen of y.

00:33 So we know by a logarithmic rule that we can say, we can change that into something that says ln of x equals ln of y raised to the fifth power.

00:45 Because that's one of, that's a logarithmic rule that when we have a number that is multiplied with a logarithm and with any variable or any number inside, we can just say that the number can be used as an exponent for the variable inside or the number inside.

01:06 So we have l .n of x equals ellen of y to the fifth.

01:10 And another rule that we need to make sure that we know is if we have an equation that says l of a certain number, or let's just say a here, is equal to ellen of b, then a has to equal b.

01:25 So this would mean that x is equal to y to the fifth because that way we have our first equation transformed into a equation with only variables instead of logarithms and now let's work with the second equation right there so there we have log base 2 of x is equal to 3 plus 2 times log base 2 of y so we know that we can tell here that the log base two are similar or they're the same in this case with logarithms with the same base and we can simply just put them in the same side whenever we have something with a logarithm um term being added with anything else we can just take the logarithms that have the same bases onto the same side.

02:30 So we can say 3 is equal to log base 2 of x minus 2 times log base 2 of y.

02:40 Let's just simplify it even more by using the previous rule of exponents that we just discussed...

Please give Ace some feedback