Solving Exponential Equations with Logarithms

The video tutorial explains how to solve exponential equations using logarithms and common bases. It begins with a simple example of solving 3^x = 5 using logs and progresses to more complex equations like 5^(2x+3) = 8 and 3^(x+2) = 4^(2-x). The tutorial also covers solving equations without logs by using common bases, demonstrated with examples like 27^(x+1) = 9^(3-x) and 16^(5x-2) = 32^(3x+1). Each example is broken down step-by-step to ensure understanding.

What is the first step in solving an equation where x is an exponent using a calculator?

• Subtract the base from both sides

• Divide both sides by the base

• Take the logarithm of both sides

• Multiply both sides by the exponent

What property of logarithms allows you to move the exponent to the front?

• Logarithm quotient rule

• Logarithm product rule

• Logarithm base change rule

• Logarithm power rule

How do you isolate x in the equation log(3^x) = log(5)?

• Multiply both sides by x

• Add log(3) to both sides

• Subtract log(5) from both sides

• Divide both sides by log(3)

In the equation 5^(2x+3) = 8, what should you do after taking the logarithm of both sides?

• Distribute the log(5)

• Divide both sides by log(5)

• Subtract 3 from both sides

• Multiply both sides by 2

What is the result of log(8) / log(5) in the context of solving logarithmic equations?

• Approximately 1.29203

• Approximately 2.0794

• Approximately 1.6094

• Approximately 0.465

When can you solve exponential equations without using logarithms?

• When the exponents are negative

• When the equation only involves multiplication

• When the equation involves addition

• When the bases can be made common

How do you simplify the equation 27^(x+1) = 9^(3-x) using common bases?

• Multiply both sides by x

• Convert 27 and 9 to powers of 3

• Divide both sides by 9

• Convert 27 and 9 to powers of 9

What is the correct value of x for the equation 4^(x+2) = 8^(3x-1) when simplified using common bases?

• x = 1/2

• x = 3/5

• x = 2

• x = 1

What is the base conversion for 16 and 32 in the context of solving exponential equations?

• Both to base 8

• Both to base 16

• Both to base 4

• Both to base 2

What is the final value of x in the equation 16^(5x-2) = 32^(3x+1)?

• x = 2/5

• x = 5/13

• x = 5/2

• x = 13/5