Exponential and Logartihmic Functions

Unit 5: Exponential and Logarithmic Functions Essay

Exponential Function

Exponential Functions:

An exponential equation is a type of transcendental equation, or equation that can be solved for one factor in terms of another. An exponential function f with base a is denoted by f (x) = ax, where a is greater than 0, a can not equal 1, and x is any real number. The base 1 is excluded because 1 to any power yields 1. For example, 1 to the fourth power is 1×1×1×1, which equals 1. That is a constant function which is not exponential, so 1 is not allowed to be the base of an exponential equation. Otherwise, the base of a can be any number that is greater than 0 and isn’t 1, and x can be any real number. The equation for the parent function of an exponential functions follows as so:

-Domain: (-, )

-Range: (0, )

-Intercept: (0,1)

-It increases

-x-axis is the horizontal asymptote

-Continous

-At x=1, y=a.

This equation shifts from the parent function based on the equation f(x) = k+a(x-h) . In this equation, k shifts the parent function vertically, up or down, depending on the value of k. The h value shifts the parent function to the left or right. If h equals 1, it goes to the right 1 unit, if it is negative 1, it goes to the left 1 unit. If a is negative, the parent function is reflected on the x-axis. If x is negative, the parent function is reflected on the y-axis.

In many applications, the natural base e is the most convenient base in an exponential equation. The value e is approximately 2.718281828. The natural base e works exactly like any other base. It is easy to think of e as a substitution for a in f (x) = ax. Its graph looks as so:

-Domain: (-, )

-Range: (0, )

-Intercept: (0,1)

-It increases

-x-a...

... middle of paper ...

... Association, if a the sound of a plane taking off is 1,000,000,000,000 times the threshold sound, and if the sound of a hand drill is 10,000,000,000 times the threshold sound, during which sound would you wear hearing protection?

Db(plane) = 10log[ I ÷ I0 ]

= 10log[ (1,000,000,000,000)I0 ÷ I0 ]

= 10log[1,000,000,000,000]

= 120 Decibels

Db(hand drill) = 10log[ I ÷ I0 ]

= 10log[ (10,000,000,000)I0 ÷ I0 ]

= 10log[10,000,000,000]

= 100 Decibels

Conclusion: If I am near a plane taking off, I should wear ear protection, and if I am near a hand drill that is on, I don’t need ear protection. This is because a plane taking off is 120 decibels, and the sound of a hand drill is 100 decibels, and 120 decibels is greater than 110 decibels, so you need protection, and 100 decibels is less than 110 decibels, so you don’t need ear protection.