Solving Equations with Logarithms

The video tutorial explains how to solve the equation 3^x = 2^(x+1) by using natural logarithms. It begins with an introduction to the problem and proceeds to apply logarithms on both sides of the equation. The power rule for logarithms is used to simplify the equation, allowing for the isolation of x. Finally, the solution is derived by dividing both sides, resulting in x = ln(2) / (ln(3) - ln(2)).

What is the initial equation that needs to be solved in this problem?

• 3^x = 2^(x+1)

• 2^x = 3^(x+1)

• x^3 = 2^(x+1)

• 3^(x+1) = 2^x

Which mathematical operation is suggested to solve the equation?

• Adding 1 to both sides

• Multiplying both sides by x

• Using logarithms

• Taking the square root

What type of logarithm is used in the solution process?

• Common logarithm

• Binary logarithm

• Decimal logarithm

• Natural logarithm

What is the purpose of using logarithms in this problem?

• To simplify the equation

• To eliminate the variable

• To convert the equation to a polynomial

• To find the derivative

What rule is applied to bring the exponent in front of the logarithm?

• Quotient rule

• Power rule

• Chain rule

• Product rule

After applying the power rule, what expression represents the left side of the equation?

• ln(x) * 2

• ln(x) * 3

• x * ln(3)

• x * ln(2)

What is the next step after applying the power rule to both sides of the equation?

• Factor out x

• Distribute the logarithm

• Subtract x from both sides

• Add x to both sides

What happens to the x terms after distributing the logarithm?

• They are subtracted from each other

• They are added together

• They are divided by each other

• They are multiplied by each other

What operation is performed to isolate all x terms on one side of the equation?

• Division

• Multiplication

• Subtraction

• Addition

What is the expression obtained after factoring out x?

• x(ln(2) + ln(3))

• x(ln(3) + ln(2))

• x(ln(3) - ln(2))

• x(ln(2) - ln(3))