Solve log4(x-1)=log2(x-3).

Name: Problem 4, Chapter 1: Mathematics for Joint Entrance Examination JEE (Advanced) Trigonometry
Uploaded: 2021-07-05T17:35:35-08:00
Duration: 3 min 28 s
Channel: Chris Wojturski
Description: Solve log4(x-1)=log2(x-3).

Key Concepts

Logarithmic Functions

A logarithmic function is the inverse of an exponential function. It helps in solving equations where the variable is in the exponent, and its properties are fundamental for understanding relationships involving exponential growth or decay. The function log_b(a) answers the question 'to what power must the base b be raised to produce a?'

Change of Base Formula

The change of base formula allows you to rewrite logarithms in terms of logarithms with a different base, facilitating comparison or calculation. This formula is particularly useful when dealing with logarithms of different bases, as it converts them into a common base, making equations easier to solve.

Domain Restrictions for Logarithms

Logarithmic functions are only defined for positive arguments. This means that when solving equations involving logs, one must ensure that the expressions inside the logarithms are greater than zero. Recognizing these domain restrictions is crucial for determining valid solutions in any logarithmic equation.

Solving Logarithmic Equations

Solving logarithmic equations generally involves applying logarithmic properties, the change of base formula, and considering domain restrictions to isolate and determine the variable. This process often includes rewriting logarithms in a common base or converting the logarithmic form into its equivalent exponential form.

Transcript

00:03 All right.

00:04 Here in 94, we have a lovely review problem.

00:07 We want to solve a logarithmic equation.

00:11 So you're going to have to remember a couple of things.

00:12 You're going to have to remember some properties of logarithms and how to solve a rational equation, actually.

00:21 So without further ado, let's go to it.

00:24 But the logarithmic property that applies here, of course, is that if you have the log of, a log, of a, i should say, difference of two logarithms, log a minus log b, you can condense that into the expression log of a over b.

00:42 So that's what we're gonna do here with those expressions x plus three and x minus one.

00:49 So log base four of x plus three minus log base four of x minus one can be written as log base four of x plus three over.

01:03 X minus 1 and that equals 2 and now we convert this to exponential form remember how to convert a logarithm into its equivalent exponential expression here's your oops didn't mean to scribble it out i meant to highlight it here's your base and here's your exponent and the expression x plus 3 over x minus 1 is what that has to equal.

01:36 So using the conversion directly, that would be 4 to the second, has to be equal to x plus 3 over x minus 1.

01:45 That's the equivalent conversion out of logarithmic form.

01:49 I'm going to rewrite a little bit down here...

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