Exploring Exponential and Logarithmic Equations

This lesson covers solving exponential and logarithmic equations. It begins with an introduction to the properties of exponents and logarithms, followed by four examples. The first example demonstrates solving exponential equations using common bases. The second example shows how to rewrite equations using logarithms. The third example involves using logarithmic properties when bases are unrelated. The final example focuses on solving logarithmic equations using natural logs and addresses extraneous solutions.

What is the solution to the equation when one half raised to the power of (x+7) equals four raised to the power of (3x)?

• x = 2

• x = -1

• x = 1

• x = 0

How can you express the equation 17 equals 4 to the power x using logarithms?

• log base 17 of 4 equals x

• log base x of 17 equals 4

• log base x of 4 equals 17

• log base 4 of 17 equals x

What is the approximate decimal value when solving the equation 17 equals 4 to the power x using logarithms?

• 4.78

• 1.23

• 2.04

• 3.56

How do you handle equations with bases of 3 and 5 using logarithms?

• Add the bases together

• Multiply both sides by the base

• Convert bases to a common number

• Take the log of both sides

What is the result of factoring out x in the equation involving logs of base 3 and 5?

• log of 3 plus log of 5

• log of 5 minus log of 3

• log of 5 plus log of 3

• log of 3 minus log of 5

What is the final value of x when solving the equation with logs of base 3 and 5?

• 2.15

• 3.15

• 1.15

• 4.15

What property allows us to equate the expressions inside the logarithms when they are equal?

• Distributive property

• Additive property

• Multiplicative property

• One-to-one property

What is the correct solution for the logarithmic equation involving x squared minus 16 and 6x?

• x = 6

• x = -2

• x = 8

• x = 16

Why is x = -2 considered an extraneous solution in the logarithmic equation?

• Logarithm of a negative number is undefined

• Negative solutions are not allowed

• It does not satisfy the original equation

• The equation simplifies to a positive number

What should be checked after solving logarithmic equations to ensure solutions are valid?

• All of the above

• Substitute back into the original equation

• Verify solutions using a calculator

• Check for extraneous solutions