Exponential and Logarithmic Functions

What Is an Exponential Function?

If we take a look at a linear function its change is always constant, you can expect the value of ‘x’ at a point similar to the value of ‘x’ at any other point. This is because it is nothing else but the slope of the given equation.

In a linear equation, if one moves one unit on the X-axis, the difference remains the same because the slope is the same for all the values of ‘x’.

But, in an exponential function, the value of ‘x’ is the exponent itself. An example of an exponential function is y = 2^x.

If we put the value of x = 0 in the above-given equation, then the value of y becomes 0. Similarly,

If x = 1 then y = 2

If x = 2 then y = 4

If x = 3 then y = 8

If x = 4 then y = 16

If x = 5 then y = 32

If x = 6 then y = 64

And so on.

We can notice that if we increase the value of x, the value of y also increases. The graph for an exponential function thus increases gradually.

Here is a difference between a Quadratic and Exponential function-

Quadratic Function:

y = x² + 4x

Exponential Function:

y = 2^x

The above quadratic equation can easily be made equal to zero but putting the values of ’x’ as 0 in ‘x2 + 4x’ however, an exponential function can never be equal to zero. The value can get very close to zero but not zero.

Note that a negative value of ‘x’ in an exponential function will result in a fractional value with the numerator always as one.

Logarithmic Functions

A logarithmic function is one that is obtained from exponential functions. Logarithmic functions can calculate exponent values that are not in an integer form.

Finding the value of ‘x’ in an exponential expression-

4x = 8

Therefore, the value of x in the above equation is 2. This is easy, but finding the value of x in 4x = 10 becomes complex. This is when logarithmic functions come into action. We now convert the exponential equation into a logarithmic equation by following certain logarithmic rules. We write 4x =  10 as log410 = x. This is how we find the value of x by looking up for values of the given log number in log calculators or logbooks.

Here is a difference between an exponential and logarithmic function-

Exponential function: y = 2x , Logarithmic Function: log210 = x.

The base of 10 aids in calculating logarithmic values. ‘Napier Logarithm Table helps in finding logarithmic value. Decimals, fractions, positive whole numbers can refer to the logarithmic table but the value cannot be calculated for values that are negative.

Here are some important properties related to logarithmic functions-

• logab = loga + logb
• loga/b = loga – logb
• logba = loga/logb
• log ax = x loga
• log1a = 0
• logaa = 1

Examples of logarithmic functions-

Exponential function	Logarithmic function	How to read the function –
2x = 10	Log210 = x	Log base 2 of 10 is x.
104 = 10000	Log 10000 = 4	Log base 10 of 10000 is 4 .
72 = 49	Log749 = 2	Log base 7 of 49 is 2.
53 = 125	Log5125 = 3	Log base 5 of 125 is 3.
ay = x	Logax = y	Log base ‘a’ of ‘x’ is ‘y’.

Conclusion

Logarithmic and exponential functions play an important function in our day-to-day lives. Logarithmic and exponential functions are used to find exponential equations.

They also help us to find and note down readings of earthquakes, on a ‘Richter Scale’. Sound measurement in decibel scale is also done by using the properties of logarithmic functions.

The brightness of a star and pH value of the acidic and alkaline nature of solutions in chemistry is also based on logarithmic functions.

This guide will surely enable you to reduce problems and make it easy for students to perform logarithmic and exponential function solutions.