Solving Exponential and Logarithmic Equations

This video tutorial covers chapter 3.4, focusing on solving exponential and logarithmic equations. It introduces key strategies like the one-to-one and inverse properties, and provides multiple examples to illustrate solving techniques. The video concludes with a story problem involving continuous compounding interest.

What is the one-to-one property for exponential equations?

• If log_a(x) = log_b(y), then x = y

• If a^x = b^y, then x = y

• If a^x = a^y, then x = y

• If a^x = a^y, then x ≠ y

How can you solve the equation 2^x = 512 using the one-to-one property?

• Rewrite 512 as 2^8 and set x = 8

• Rewrite 512 as 2^9 and set x = 9

• Rewrite 512 as 2^10 and set x = 10

• Rewrite 512 as 2^7 and set x = 7

In Example 2, what is the first step to solve 4 * 3^x = 64?

• Divide both sides by 4

• Multiply both sides by 4

• Add 4 to both sides

• Subtract 4 from both sides

What is the solution to the equation e^x = 1?

• x = 1

• x = e

• x = 0

• x = -1

How do you solve the equation 6 * 2^(t+5) + 4 = 11?

• Add 4 to both sides, then multiply by 6

• Subtract 4 from both sides, then divide by 6

• Add 4 to both sides, then divide by 6

• Subtract 4 from both sides, then multiply by 6

What is the first step in solving e^(2x) - 7e^x + 12 = 0?

• Multiply both sides by e

• Take the natural log of both sides

• Divide both sides by e

• Recognize it as a quadratic in form

How do you solve the equation ln(x) = 2/3?

• Take the natural log of both sides

• Multiply both sides by 2/3

• Exponentiate both sides to get x = e^(2/3)

• Divide both sides by 2/3

What is the solution to the equation log(x) + log(x-9) = 1?

• x = 0

• x = -1

• x = 9

• x = 10

How long will it take to double an investment of $500 at 6.75% interest compounded continuously?

• Approximately 10.3 years

• Approximately 15 years

• Approximately 5 years

• Approximately 20 years