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

Name: Problem 53, Chapter 12: Algebra and Trigonometry
Uploaded: 2021-07-29T13:22:01-08:00
Duration: 3 min 5 s
Channel: Ziya Ogron
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...

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Key Concepts

Properties of Logarithms

This concept involves the basic rules that govern logarithmic functions, such as the product, quotient, and power rules. These rules allow us to simplify and manipulate logarithmic expressions, which is crucial when solving equations that involve logarithms by transforming expressions and isolating variables.

Exponential and Logarithmic Equations

Understanding how to convert between exponential and logarithmic forms is key when solving equations with logarithms. This process usually involves rewriting a logarithmic equation in its equivalent exponential form so that the variable can be isolated and solved for.

Systems of Equations

This refers to solving multiple equations that share common variables. In the context of logarithmic equations, this may involve using substitution or elimination methods to reduce the system to simpler forms, ultimately finding the values that satisfy all equations simultaneously.

Domain Restrictions in Logarithmic Functions

An important aspect when working with logarithmic equations is recognizing that the arguments of logarithms must be positive. This restriction must be considered throughout the solution process to ensure that all proposed solutions are valid within the domain of the logarithmic functions.

Transcript

00:01 In this problem, we're asked to solve this system of equations using any method.

00:06 So we know that if we have a number in front of natural log, this number is the exponent for the y.

00:13 So therefore, i can rewrite this equation as natural log of x equals to natural log y to the 4.

00:21 So because both of them have natural log, then we know that this is the same thing as x equals to y to the 4.

00:30 And then for the second equation, we need to put the logs on the same side of the equation.

00:37 So therefore, that's going to become log of 3x, subtract 2 log 3 of y equals to 2.

00:49 And then we can condense this logarithm because we know minus means dividing.

00:54 So this is the same thing as log base.

00:56 Before we do that, same thing.

00:58 The 2 in front of log is going to be the exponent.

01:03 So this can be rewritten as log base 3 of x minus log base 3 of y squared equals to 2.

01:14 So now we can condense it.

01:16 Minus means that you're going to divide them.

01:19 So this is the same thing as log base 3 of x over y squared equals to 2.

01:27 So now i'm going to convert this logarithm into exponential form.

01:32 So your base is 3.

01:34 And so therefore, this is the same thing as 3 squared equals to x over y squared.

01:42 So now three squared is nine...

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