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

Solve the logarithmic equation and eliminate any extraneous...

Key Concepts

Extraneous Solutions

Extraneous solutions are results produced during the algebraic manipulation of an equation that do not satisfy either the original equation or the associated domain conditions. In the context of logarithmic equations, checking potential solutions against the domain restrictions is crucial to avoid including invalid answers.

Domain Restrictions for Logarithms

When working with logarithmic functions, it is critical to ensure that the arguments of all logarithms are positive since logarithms are undefined for zero or negative values. Determining the domain helps in identifying valid solutions and eliminating any that do not satisfy the necessary conditions.

Logarithm Properties

These include fundamental rules such as the product rule, which allows the combination of two logarithms into one by expressing the sum as the logarithm of the product of the two arguments. This property is essential in simplifying logarithmic equations and transforming them into a form that is easier to solve.

Transcript

00:01 To solve the question, we need to use the addition property of logarithm.

00:06 When the bases are same, we know that we can change addition to multiplication or multiplication to addition.

00:14 So let me use the property here, logarithm x times 2x minus 1 equals 1.

00:25 So we change operation, addition to multiplication, and it will be logarithm x times 2x squared, and x times minus 1 equals 1.

00:45 And to get x, you need to change a question from logarithmic form to exponential form.

00:52 The base is not written.

00:54 It's common logarithm, so it is 10.

00:57 We know that for exponential form 10 to the first power equal stuff inside of logarithm.

01:06 So it will be 2x squared minus x equal 10 to the first power.

01:12 That is 10.

01:14 And let's make left side 0.

01:18 So we will add negative 10 both sides.

01:23 Then we will get 2x square minus x and minus x.

01:27 N minus 10 equal 0.

01:31 And then let's use quadratic formula to get x solution to solve the question.

01:40 So x equals minus b.

01:45 Remember the quadratic formula plus minus root of b square.

01:53 B square is minus 1 square minus.

01:57 4 times, sorry, here.

02:03 4, ac 2 and negative 10 in a square root over 2 times a is 2.

02:15 So it will be x equals plus 1 plus minus.

02:24 This will be 1 plus 80 in wood.

02:31 4 equals 1 plus minus root of 81 over 4 so it is root of 81 it is 9 so 1 plus minus 9 over 4 then one of the root will be 1 plus 9 over 4 that is 10 over 4 it's 5 over 2 and the other 1 is 1 minus 9 over 4.

03:05 It is minus 8 over 4 equals minus 2...

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