Exponential and Logarithmic Functions

We examine the significance and relevance of the number e and utilize the features of these Exponential and Logarithmic Functions to solve equations with exponential or logarithmic components. Hyperbolic and inverse hyperbolic functions, which combine exponential and logarithmic functions, are commonly defined. We also defined the essential number e in this article, which is the basis for the natural logarithm and the standard base for exponential functions in calculus. Navigators, scientists, engineers, surveyors, and others quickly embraced exponential and logarithmic functions to make high-accuracy computations easier. Logarithm tables can be used to replace lengthy multi-digit multiplication processes with table look-ups and easier addition.

Evaluating Exponential Functions

Recall the properties of exponents: If  x  is a positive integer, then we define  bx=b⋅b⋯b  (with  x  factors of  b ). If x is a negative integer, then  x=−y  for some positive integer y , and we define  bx=b−y=1/by . Also,  b0  is defined to be  1 . If  x  is a rational number, then  x=p/q , where  p  and  q  are integers 

Rule: Laws of Exponents

For any constants  a>0 , b>0 , and for all x and y,

• bx⋅by=bx+y 
• bx/by=bx-y  
• bx⋅by=bx+y 
• (ab)x=ax⋅bx
• ax⋅bx=(ab)x

Exponential and Logarithmic Functions

An exponential equation has a variable that is expressed as an exponent. A logarithmic equation is one in which the logarithm of an expression with a variable is used. To answer exponential equations, check to determine if both sides of the equation can be written as powers of the same integer. If you can’t, take the common logarithm of both sides of the equation and use property 7 to solve the problem.

Exponential Functions

Exponential functions expand at an exponential rate—that is, at a breakneck speed. Two squared equals 4; two cubed equals 8, but by the time you get to 2 7, you’ve already gotten to 128, in four little steps from 8, and it just gets quicker from there. For example, four additional steps will increase the value to 2,048.

Any function defined by y = xb, where b > 0, b ≠ 1, and x is a real number, is called an exponential function.

All exponential functions, f ( x) = xb, b > 0, b ≠ 1, will contain the ordered pair (0, 1), since b 0 = 1 for all b ≠ 0. Exponential functions with b > 1 will have a basic shape.

The form of an exponential function is ax, where a is a constant; examples are 2x, 10x, and ex. The logarithmic functions are the inverses of exponential functions, or functions that “undo” exponential functions, like the cube root function “undoes” the cube function:( 23)3=2. It’s worth noting that the original function undoes the inverse function as well: (8–√3)3=8.

Let’s say f(x)=2x. The logarithm base 2 is the inverse of this function, abbreviated log2(x) or (particularly in computer science circles) lg (x). What exactly does this imply? The logarithm must reverse the operation of the exponential function,, the exponential function creates 23=8, and the logarithm of 8 must return us to 3.

Logarithmic Functions

We may explain the inverses of exponential functions, which are logarithmic functions, using our knowledge of exponential functions. When we need to consider a phenomenon that has a large range of values, such as pH in chemistry or dB in sound levels, we can use them.

The exponential function  f(x)=bx  is one-to-one, with domain  (−∞,∞)  and range  (0,∞) . Therefore, it has an inverse function, called the logarithmic function with base b. For any  b>0,b≠1, the logarithmic function with base b, denoted  logb, has a domain  (0,∞)  and range  (−∞,∞) , and satisfies

logb(x)=y  if and only if  by=x.

An exponent is a logarithm. In logarithmic form, every exponential expression may be recast. If 8 = 23, for example, the base is 2, the exponent is 3, and the result is 8. 

Notice how the numbers have been rearranged.

Exponents v/s Logarithms

An exponent is just a means to express that a number has been multiplied several times. In the equation, 32 = 3*3 = 9, the base of the exponent is 3 and the superscripted 2 is the exponent or power. An exponential function indicates how many times the base should be multiplied by itself. Some instances are as follows:

53 = 5*5*5 = 25*5 =125 means taking the base 5 and multiplying it by itself three times.

• This is how an exponential function is written: b is the base, and x is the exponent.
• The inverse of an exponential function is logarithmic or log function.

1. Exponential function form: 32 = 9
2. Logarithmic function form: log base 3 of 9 = 2

Take a moment to examine both forms. In exponential function form, the solution is 9. The 2 signifies the exponent and is the answer in log form. What were we talking about when we said it was a log? A log is an exponent or log = exponent in another format.

The decimal or common logarithm is a logarithm with base 10 (that is, b = 10) that is extensively used in science and engineering. Because of its easier integral and derivative, the natural logarithm takes the number e (that is, b 2.718) as its basis. It is widely used in mathematics and science.

Logarithmic scales condense large quantities into smaller units. The decibel (dB), for example, is a measure for expressing ratios as logarithms, primarily for signal strength and amplitude (of which sound pressure is a common example). pH is a logarithmic measure of an aqueous solution’s acidity in chemistry. In scientific equations and measures of the complexity of algorithms and geometric shapes known as fractals, logarithms are commonly used.

Product, quotient, power, and root

The total of the logarithms of the numbers being multiplied is the logarithm of the product; the difference of the logarithms is the logarithm of the ratio of two integers. The logarithm of a number’s p-th power is p times the number’s logarithm; the logarithm of a number’s p-th root is the number’s logarithm divided by p. These identities are listed in the table below, along with examples. Each of the identities can be derived after substitution of the logarithm definitions .

Conclusion

Exponential and logarithmic functions are the inverses of one another. We look at the features of the exponential and logarithmic functions, as well as their graphs and the rules for manipulating exponents and logs, to see how they are related.