Solving Logarithmic Equations: Key Techniques and Examples

Why is it necessary to check for extraneous solutions in logarithmic equations?

• Because logarithms can only be applied to positive numbers

• To ensure the solution satisfies the original equation

• Because all logarithmic equations have extraneous solutions

• To make the equation easier to solve

How can you combine 2 * log base 3 of x into a single logarithmic term?

• log base 3 of x^2

• log base 3 of 2x

• log base 3 of x / 2

• 2 / log base 3 of x

How do you rewrite 2 * log base 3 of x as a single logarithm?

• By adding the coefficient to the base

• By dividing the coefficient by the base

• By moving the coefficient as an exponent

• By subtracting the coefficient from the base

What does subtracting logarithmic terms with like bases result in?

• Addition

• Division

• Multiplication

• Subtraction

After converting the logarithmic equation to an exponential form, what is the next step?

• Multiply both sides by x + 6

• Factor the quadratic equation

• Check for extraneous solutions

• Subtract 3x from both sides

What property allows the conversion of a logarithmic equation to an exponential form?

• The property of exponents

• The property of logarithms

• The distributive property

• The associative property

Which of the following is a correct factor of the quadratic equation derived?

• x - 3 and x + 6

• x - 6 and x + 3

• x + 6 and x - 3

• x^2 - 3x + 18

What are the potential solutions obtained after factoring the quadratic equation?

• x = 6 and x = -3

• x = 3 and x = -6

• x = -6 and x = -3

• x = 6 and x = 3

Why is x = -3 considered an extraneous solution?

• It does not satisfy the original equation

• It results in a negative number inside the logarithm

• It makes the logarithm undefined

• It is not a real number

What is the final solution for the logarithmic equation?

• No solution

• x = 3

• x = -3

• x = 6