Logarithmic and Quadratic Equations

The video tutorial explains how to solve a logarithmic equation involving the square of log base three of y and log base three of y raised to the power of negative two. It demonstrates the application of the power property of logarithms, substitution to simplify the equation, and solving the resulting quadratic equation by factoring. The tutorial concludes by converting the logarithmic equations to exponential form to find the solutions, which are y equals one over two hundred forty-three and y equals twenty-seven.

Why can't the logarithm be isolated directly in the given equation?

• Because the equation is already in its simplest form.

• Because the logarithm is squared and also appears with a negative exponent.

• Because the base of the logarithm is not specified.

• Because the equation involves multiple variables.

What property of logarithms is applied to transform the equation initially?

• Change of Base Property

• Quotient Property

• Power Property

• Product Property

What substitution is made to simplify the equation into a quadratic form?

• u = 3 raised to the power of y

• u = y

• u = log base 3 of y

• u = y squared

What are the factors of the quadratic equation after substitution?

• u + 5 and u - 3

• u + 7 and u - 2

• u + 2 and u - 7

• u + 3 and u - 5

What is the next step after factoring the quadratic equation?

• Solve for u by setting each factor to zero.

• Multiply the factors back together.

• Add a constant to both sides.

• Divide both sides by a common factor.

How is the logarithmic equation converted to an exponential form?

• By using the base as the result and the exponent as the base.

• By using the base as the exponent and the result as the base.

• By using the result as the base and the base as the exponent.

• By using the base as the base, the exponent as the exponent, and the result as the result.

What is the value of y when log base 3 of y equals negative five?

• y = 1/27

• y = 243

• y = 1/243

• y = 27

What is the value of y when log base 3 of y equals three?

• y = 1/27

• y = 27

• y = 81

• y = 9

What are the final solutions for y in the given equation?

• y = 1/9 and y = 3

• y = 1/243 and y = 27

• y = 243 and y = 1/27

• y = 9 and y = 81

What is the significance of converting logarithmic equations to exponential form?

• It changes the base of the logarithm.

• It allows for easier calculation of the variable.

• It simplifies the equation to a linear form.

• It eliminates the need for substitution.