Logarithmic Equations and Change of Base Formula

The video tutorial explains how to solve the exponential equation 6^x = 45 by using logarithms. The instructor demonstrates taking the logarithm of both sides with base 10, which is convenient for calculator use. The solution is calculated and rounded to the nearest hundredth, resulting in approximately 2.12. The video also verifies the solution by raising 6 to the power of 2.12 and discusses the precision of the result. Finally, the tutorial explains the change of base formula, showing how it can be applied to solve the equation using different logarithmic bases.

What is the initial equation that needs to be solved?

• 6^x = 45

• x^6 = 45

• 6^x = 54

• x^6 = 54

Why is log base 10 used in the solution?

• Because calculators typically have a log base 10 function

• Because it is the only logarithm available

• Because it provides an exact solution

• Because it is easier to calculate manually

What property of logarithms allows the exponent to be moved in front?

• Logarithm of a power

• Logarithm of a sum

• Logarithm of a quotient

• Logarithm of a product

What is the approximate value of x when rounded to the nearest hundredth?

• 2.10

• 2.20

• 2.12

• 2.15

How is the accuracy of the solution verified?

• By checking if 6^3 equals 45

• By checking if 6^2.12 is close to 45

• By checking if 6^2.12 equals 45

• By checking if 6^2 equals 45

What happens when more precision is added to the calculation of 6^x?

• The result becomes closer to 45

• The result becomes further from 45

• The result becomes less accurate

• The result becomes exactly 45

What is the change of base formula used for?

• To convert between different logarithm bases

• To solve quadratic equations

• To simplify fractions

• To calculate square roots

What alternative method is mentioned for solving 6^x = 45?

• Using log base 6

• Using log base 8

• Using log base 2

• Using log base 4

What does the left-hand side of the equation simplify to when using log base 6?

• x

• 1

• 6

• 45

What is the final expression for x using the change of base formula?

• log base 6 of 10 over log base 6 of 45

• log base 6 of 45 over log base 6 of 10

• log base 10 of 6 over log base 10 of 45

• log base 10 of 45 over log base 10 of 6