Solve each system. Use any method you wish. { ln x=4 ln y log3 x=2+2 log3 y .

Name: Problem 53, Chapter 8: College Algebra
Uploaded: 2020-06-24T00:07:38-08:00
Duration: 5 min 1 s
Channel: Sushmit Acharya
Description: Solve each system. Use any method you wish. { ln x=4 ln y log3 x=2+2 log3 y .

Solve each system. Use any method you wish. { ln x=4 ln...

Key Concepts

Logarithm Properties

Understanding the core properties of logarithms is essential for manipulating and simplifying logarithmic equations. Key properties include the power rule (log_b(a^n) = n log_b(a)), which allows exponents to be brought in front of the logarithm, and the basic identity that if log_b(A) = log_b(B), then A = B. These properties are foundational when re-writing logarithmic equations in a more algebraically manageable form.

Conversion Between Logarithmic and Exponential Forms

Converting equations from logarithmic to exponential forms is a crucial technique when solving logarithmic equations. For example, an equation such as ln(x) = 4 ln(y) can be rewritten using the property of logarithms to yield x = y^4. This conversion makes it easier to manipulate the equations to isolate and solve for the unknown variables.

Solving Systems of Equations Using Substitution

When dealing with a system of equations, especially those involving logarithms, the substitution method is a valuable tool. By expressing one variable in terms of another using one equation (for instance, rewriting x as y^4), you can substitute this expression into the other equation. This method reduces the system to a single equation in one variable, thereby simplifying the process of solving the system.

Transcript

00:01 Alright, here we have two different, well, one system of nonlinear equations with logarithms, whether we have natural logs right here or regular logarithms here.

00:12 So the best thing to do would be just simplify these into regular equations so that we don't have to deal with these logarithms, or whether they be based with any numbers or based with e in this case.

00:25 So let's just start off with the first one.

00:27 So we have ln of x equals 4 times allen of y.

00:30 One logarithmic rule that we have to mention for this is there's a rule that says log base a of b let's say and log base a of c if the bases are the same we can say that the b and c or whatever values and there are equivalent of each other so we can say b equals c so when we have ellen of x allen of x equals for ellen of y so first let's short this out with the another logarithmic rule that says a times the logarithm of something would be equal to logarithm of that something raised to the a power.

01:16 So here we have ln of x equals ln of y to the fourth.

01:22 And based off of this rule, we can say that y to the fourth equals x.

01:29 So we have our first equation.

01:31 So now let's delve into our second equation.

01:35 So log base 3 of x equals 2 plus 2 times log base 3 of y.

01:49 So before we even make anything more simpler, let's just make sure we rewrite and get the 2 up with the y and say log base 3 x equals 2 plus log base 3 y squared.

02:06 And here, the one thing we can realize here is we have another logarithmic property.

02:18 Here, one other logarithmic rules that we have to discuss, as i said, is this one...

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