Logarithmic functions mathematics

Introduction 

The inverses of exponential functions are logarithmic functions, and an exponential function may be represented in logarithmic form. Similarly, all logarithmic functions may be expressed as exponential functions. Logarithms are extremely important in that they allow us to work with very huge quantities while altering much smaller numbers. 

We know that if 5x=5, then x=1 or if 5x=25, then x=2 or if 5x=125, then x=3. Inspection cannot answer the equation 5x=20. We could assume, however, that x is between 1 and 2. Similarly, the answer for the equation 5x=60 is between 2 and 3. To solve an exponential equation with an unknown exponent, we must employ a new function known as a logarithmic function. 

Logarithmic Functions 

If x and b are positive real numbers, such that, b1. Then, y=logbx is called the logarithmic function with base b and y=logbx is equal to by=x.

This indicates that the logarithm y is the exponent, to which b must be raised to obtain x. The expression y=logbx is called the logarithmic form of the equation, and the expression  by=x is called the exponential form of the equation. 

A logarithmic function has a tight connection with an exponential function of the same base, according to its definition. A logarithmic function is, in reality, the inverse of an exponential function. For example, the procedures below determine the inverse of the exponential function described by f(x)=bx. 

The inverses of exponential functions are logarithmic functions, and an exponential function may be represented in logarithmic form. 

A new term or symbol would have to be invented if x = 2y were to be solved for y so that it could be expressed in function form. If x = 2y, then y= (the power on base 2) to equal x. To meet this requirement, the term logarithm, abbreviated log, was established. y =log2 x is a rewrite of this equation.

The Common and Natural Logarithmic Function 

The logarithmic function with base 10 is called the common logarithmic function and is denoted by y= log x. Observe that the base is not explicitly written but is understood to be 10. That is, y= log10x is written simply as y =log x. 

When working with logarithms, the most commonly utilised bases are base 10 and base e. (The letter e denotes an irrational number with several applications in mathematics and science. The value of e is roughly 2.718281828…) Log base 10, is known as the common logarithm and is written as log, with the base not specified but assumed to be 10. Log base e, log e, is also known as the natural logarithm and is represented as ln. 

Properties of Logarithmic Functions 

Certain properties of logarithmic functions flow from the definition itself. Remember that 

y=logbx is equal to by=x for x>0,b>0 and b1. 

Some other properties of logarithmic functions are: 

Property 1:  logb1=0

Property 2: logbb=1

Property 3: logb bp=p

Property 4: blogbx=x 

Property 5: Also called the Product property: logb(xy)=logbx+logby

Property 6: Also called the Quotient property: logb(xy)=logbx-logby

Property 7: Also called the Power property: logbxp=p logbx

Examples:

1. Rewrite each exponential equation in its logarithmic form.  

1. 52=25

The base remains the base; hence, the above equation becomes 2=log5=25

1. 4-3=164

The above equation becomes -3=log4=164

1. Evaluate the following equations. 
2. log1010,000 

log1010,000 is the exponent to which 10 must be raised to obtain 10,000. 

Let y represent the value of the logarithm then, 

y = log10 10,000 

Rewrite the expression in exponential form. Therefore, 

10y=10,000

y= 4 

Therefore, log1010,000=4 

1. log51125

Similar to the previous question, we have to rewrite the equation in exponential form. First, let y represent the logarithm. 

y=log51125

5y=1125

y=-3

Therefore, log51125=-3

1. Use the properties to solve the given equations. Suppose,  all variable expressions within the logarithms are positive real numbers. Mention the properties used.
   
   

1. log88+log81

Using properties 1 and 2, 

log88+log81= 1+0=1 

1. 10log(x+2)
   
   

Using property 4, 

10log(x+2)= x+2 

Conclusion 

In a nutshell, logarithmic functions are not very difficult if you practice a lot. If you understand the concept, calculating and evaluating the equations becomes easy. Make sure you follow the study material notes on logarithmic functions to enhance your skills.