Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions

step 1 Here we can use addition property of logarithm. We know that when the bases are same, we can cha

Solve the logarithmic equation and eliminate any extraneous...

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Key Concepts

Algebraic Manipulation

Algebraic manipulation involves applying properties of logarithms and other algebraic techniques to transform the logarithmic equation into a simpler form, such as a quadratic or linear equation. This step is crucial in isolating the variable and finding potential solutions.

Checking for Extraneous Solutions

After solving a transformed equation, it is necessary to substitute back into the original equation to ensure that the solutions satisfy the domain restrictions. Extraneous solutions are those that, while mathematically derived, violate the conditions imposed by the logarithms, such as the requirement that their arguments must be positive.

Logarithm Properties

Key properties of logarithms, such as the sum of logarithms being equivalent to the logarithm of a product, are essential in simplifying and solving logarithmic equations. This property is used to combine multiple logarithmic terms into a single logarithmic expression, facilitating the solution process.

Domain Restrictions

When dealing with logarithmic functions, it is important to remember that the argument of a logarithm must be positive. This concept ensures that any solutions found during the process are valid, as values causing the argument to be zero or negative must be excluded from the final answer.

Transcript

00:01 Here we can use addition property of logarithm.

00:04 We know that when the bases are same, we can change addition to multiplication.

00:11 So here let's apply that rule.

00:15 Then logarithm x plus 1 times stuff inside of second logarithm x minus 1 equals 0.

00:24 And this will be logarithm x plus 1 times x minus 1.

00:29 It is x squared minus 1 equals 0.

00:33 And here we need to change it with exponential form.

00:37 The base is not written, but we know that it is 10.

00:40 It's common logaritium.

00:42 So let's change it with the exponential form.

00:47 In exponential form, stuff inside of logarithm equals base to the zero power, that is 10 to the zero power.

00:56 So x squared minus 1, 10 to 0 power is 1.

01:01 Then let's add plus 1.

01:04 Then it is x squared equals 2.

01:08 Let's take both side the square root.

01:11 Then x equals here plus minus root of 2.

01:16 And we know that for the domain of log item, we need to check our solution.

01:24 Here we have two solutions.

01:26 One of them is x equals x root of 2, and the other one is negative root of 2.

01:34 So let me write first full root of 2 in the original question...

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