Exponential and Logarithmic Equations

In this video, Mr. Banker explains how to solve exponential equations using the one-to-one property, which allows us to equate the exponents when the bases are the same. He provides examples, such as solving 2^x = 32, and demonstrates how to handle more complex equations that cannot be solved using this property alone. The video also covers converting exponential equations to logarithmic form and solving quadratic-like exponential equations using algebraic substitution.

What does the one-to-one property state about exponential equations with the same base?

• The bases must be subtracted.

• The bases must be multiplied.

• The exponents must be equal.

• The exponents must be added.

How can you rewrite 32 to have a base of 2?

• 2^4

• 2^5

• 2^3

• 2^6

What is the first step in solving the equation 20 * (1/2)^(x/3) = 5 using the one-to-one property?

• Multiply both sides by 20.

• Add 20 to both sides.

• Subtract 20 from both sides.

• Divide both sides by 20.

When the one-to-one property cannot be used, what method is suggested for solving exponential equations?

• Using trigonometric identities.

• Using logarithms.

• Using polynomial equations.

• Using geometric sequences.

How can the equation e^x = 5 be rewritten in logarithmic form?

• log base 5 of e = x

• log base e of 5 = x

• log base 5 of x = e

• log base x of 5 = e

What is the natural log of 5 expressed as?

• log(5)

• log base 2 of 5

• ln(5)

• log base 10 of 5

What algebraic strategy is used to solve the equation e^(2x) - 3e^x + 2 = 0?

• Using polynomial division.

• Using algebraic substitution.

• Using trigonometric identities.

• Using matrix operations.

In the equation e^(2x) - 3e^x + 2 = 0, what substitution is made to simplify the equation?

• Let W = 2e^x

• Let W = e^x

• Let W = 3e^x

• Let W = e^(2x)

After substitution, what type of equation is W^2 - 3W + 2 = 0?

• Linear equation

• Quadratic equation

• Exponential equation

• Cubic equation

What are the solutions for W in the equation W^2 - 3W + 2 = 0?

• W = 0 and W = -1

• W = 1 and W = 0

• W = 2 and W = 1

• W = 3 and W = 2