**Exponential Functions and its Inverse**

- an exponential function of the form y = bx , b > 0, b not equal 1, has

- a repeating pattern of finite differences
- a rate of change that is increasing proportional to the function for b > 1
- a rate of change that is decreasing proportional to the function for 0 < b < 1

- An exponential function of the form y = bx , b > 0, b not equal 1,

- has a domain X E R
- a range Y E R, Y > 0
- a y-intercept of 1
- has a horizontal asymptote at y = 0
- is increasing on its domain when b > 1
- is decreasing on its domain when 0 < b < 1

- The inverse of y = bx is a function that can be written as x = by.

- has a domain of X E R, x > 0
- a range of Y E R
- a x-intercept of 1
- has vertical asymptote at x = 0
- is a reflection of y = bx about the line y = x
- is increasing on its domain when b > 1
- is decreasing on its domain when 0 < b < 1

**Logarithms**

- a logarithmic function is the inverse of the exponential function
- The value of logbx is equal to the exponent to which the base, b, is raised to produce product x
- Exponnetial equations can be written in logarithmic form, and vice versa

**y = b^x <-> x = logby****y =**logb**X <-> x = b^y**

- Exponential and logarithmic functions are defined only for positive values of the base that are not equal to one. In other words, b not = 1, and x > 0.
- The logarithm of x to base 1 is only valid when x = 1, in which base y has an infinite number of solutions and is not a function.
- Common logarithms are logarithms wit ha base of 10. It is not necessary to write the base for common logarithms: logx means log base 10 x.

**Transformations of logarithmic Functions**

- The techniques for applying transformations to logarithmic functions are the same for those used for other functions:

- y = log x + c
- translate up c units if c > 0
- translate down c units if c < 0
- y = log( x – d)
- translate right d units if d > 0
- translate left d units if d < 0
- y = a log x
- stretch vertically by a factor of |a| if |a| > 1
- Compress vertically by a factor of |a| if |a| < 1
- Reflect in the x-axis if a < 0
- y = log (kx)
- compress horizontally by a factor of |1/k| if |k| < 1, k not = 0.
- Reflect in the y axis if k < 0.

- When all transformations are combined, they follow the form:

f(x) = a log[k(x-d)] + c

**Power of logarithms**

- The power of logarithms states that logbxn = n logbx for b > 0, b not = 1, x > 0, and n ER
- Any logarithm can be expressed in terms of common logarithms using the change of base formula:

**log base b m = log m / log b, b > 0, b not = 1, m > 0**

**Exponential Functions**

- Exponential functions and expressions can be expressed in different ways by changing the base
- Changing the base of one or more exponential expressions is a useful technique for solving exponential equations

**Solving Exponential Functions**

- An equation maintains balance when the common logarithm is applied to both sides
- The power of logarithms is a useful tool for solving a variable that appears as a part of an exponent
- When a quadratic equation is obtained, methods such as factoring and applying the quadratic formula may be useful.
- Some algebraic methods of solving exponential functions lead to
**extraneous roots,**which are not valid solutions to the original equation

**Laws of logarithms**

- The product law of logarithms states that logbX + logb Y = logb(XY) for b > 0, b not = 1, x > 0, y > 0
- The quotient law of logarithms states that logbX – logbY = logb(X/Y) for b > 0, b not = 1, x > 0, y > 0

**Applications of Exponential/ Logarithmic Functions**

**Interest rates:**

**A = (i + 1)**^{t}

Where **A** is the amount, **i** is the interest, and **t** is the time.

**Population**

**P = A**_{1}** (i)**^{t}

Where P is the population, A_{1}is the initial amount, i is the amount increase, and t is time.

**Half Life:**

**A**_{o }**= A**_{i }**(1/2)**^{t/h}

Where A_{o} is the final amount, A_{i} is the initial amount, t is time, and h is the time interval

**pH Levels:**

**pH = -log (H**^{+}**)**

Where **H**^{+}** **is the concentration of hydrogen ions

Every Integer increment in pH is 10 times more acidic

**Sound Intensity:**

**L = 10log(I / I**_{o}**)**

Where **L **is the loudness, I is the intensity, and I_{o} is the sound that is barely audible

Every Integer increment in L, decibels, is 10 times more intense

**Earthquakes:**

**M = log(E/ I**_{o}**)**

Where **M **is the Richter Number, E is the earthquake’s intensity, and I_{o} is the intensity of a referenced earthquake.

Every Integer increment in M, Richter readings, is 10 times greater the earthquake intensity.