# Exponential and Logarithmic Functions Alvaro Hurtado

A function is a relation in which each element of the domain is paired with exactly one element in the range. Two types of functions are the exponential functions and the logarithmic functions. Exponential functions are the functions in the form of y = ax, where ”a” is a positive real number, greater than zero and not equal to one.

Logarithmic functions are the inverse of exponential functions, y = loga x, where ”a” is greater to zero and not equal to one. These functions have certain differences as well as similarities between them. Also they are very useful for various situations in life.

Logarithmic functions are fairly different from the exponential functions. The first difference that we can find between them is in the equations, they are inverse to each other. The logarithmic equation is y = loga x and the exponential equation is y = ax. We can also see that the natural exponential function is different form the natural logarithmic function. The natural exponential function is y = f(x) = ex and the natural logarithmic function is f(x) = loge x = lnx , where x > 0.

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Also we can see that to graph and exponential function it always has to pass through the point (0,1).

However, both of these functions also have similarities. Both of the functions contain an ”a” which has to be greater than zero and less than one.

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Also when we graph both of the functions we can see that they will never touch an axis because of the rule that ”a” is greater than zero and less than one. To solve exponential functions, you use the same rules set of rules that you use to solve logarithmic functions. (logamn = loga m + loga n, logam / n = loga m – loga n, loga m = p x loga m, If loga m = loga n, then m = n. Where m and n are positive number, b is any positive number rather than one and p is any real number.) Also both of the functions have a base which is the ”a.” This are some of the similarities that we can see from the equations.

Logarithmic and exponential functions are very useful for many situations in real life. Exponential functions are used to estimate and graph topics that have to do with growth or any type of data that deals with an increase. For example is used to describe and graph the population of a country and its rapid change. Also it can be used to explain the exponential growth of almost anything. Logarithmic functions are also useful to calculate the interests you gain in a bank. Both of these functions are most commonly applied in finding the interest earned on an investment, population growth and carbon dating.

In conclusion we can see that even though the logarithmic and exponential functions are the inverse of each other they have similarities as well as differences between them. Also that the functions form a very important part of life because they are useful in different situations. As well, they are not only useful in some themes or topics like only biology and math but in a whole different variety, such as financial planning, banking, constructing, business and many more.

Exponential functions are functions where f(x) = ax + B where a is any real constant and B is any expression. For example, f(x) = e-x – 1 is an exponential function.

To graph exponential functions, remember that unless they are transformed, the graph will always pass through (0, 1) and will approach, but not touch or cross the x-axis. Example:

## Problem: Graph f(x) = 2x.

Logarithmic functions are the inverse of exponential functions. For example, the inverse of y = ax is y = logax, which is the same as x = ay.

(Logarithms written without a base are understood to be base 10.)

This definition is explained by knowing how to convert exponential equations to logarithmic form, and logarithmic equations to exponential form. Examples:

## Problem: Convert to logarithmic form:

8 = 2x

Logarithmic and exponential functions are used and are very helpful in many situations in life. For example exponential functions are used in biology to express the rapid growth of bacteria and understand how the radioactive decay can be used to date artifacts found. Also it can be used to explain the exponential growth of almost anything. Logarithmic functions are also useful to calculate the interests in a bank. Both of these functions are helpful to find different type of information related on bank accounts, interests, population and anything related to growth.

In conclusion we can see that logarithmic and exponential functions are relevant for different situations in life. Also that they are similar in some ways even though they are the inverse of each other. But also they have differences between them.

## Science

While studying exponential functions students can express the rapid growth of bacteria with an exponential function and understand how radioactive decay can be used to date artifacts found at archeological sites.

## Social Studies

While studying exponential growth, students can use an exponential model to express population growth.

## Technology

While studying logarithmic and exponential functions, students can explore changes in the functions and corresponding changes in their graphs. Students can also employ technology to fit a mathematical model to exponential or logarithmic data. 