Exponential Equation Formula

Exponential Equation Formula

Exponents are used in exponential equations, as the name suggests. It is known that a number’s exponent (or base) tells about how many times the original number has been multiplied. However, it is also important to know what occurs if a number’s power is a variable. An Exponential Equation Formula one in which the power is a variable and a part of the equation. To solve the exponential equations, it is needed to use the relationship between exponents and logarithms.

Exponential equations should be defined and students should understand how to solve them when the bases are the same and when they are not. They must also peruse some examples that have been solved and some practice problems. Although it can be extended to complex numbers or adapted to other mathematical objects like matrices or Lie algebras, the term typically refers to the positive-valued function of a real variable. The idea of exponentiation, or repeated multiplication, is where the exponential function got its start, but more recent definitions—there are several that are equivalent—allow it to be rigorously extended to all real arguments, including irrational values. Mathematician Walter Rudin claimed that the exponential function is “the most important function in Mathematics” due to its widespread use in both pure and applied Mathematics.

What are Exponential Equations?

An Exponential Equation Formula is one with exponential expressions, or powers with the unknowable factor x in their exponent expressions.

Without using logarithms, students will learn to solve the exponential equations in this section. In order to equalise the exponents, this method of resolution involves bringing all of the exponentials to the same base.

For instance:

32x = 36

Clearly, x must take the value 3 in order for the equality to hold true.

It is expected that students factorise, write all the numbers as powers, apply the properties of powers, and write the roots as powers in order to produce this kind of expression. They also need to occasionally change a variable to transform the equation into a quadratic one.

Types of Exponential Equations

Now that the definition of the Exponential Equation Formula has been provided and described, students can discuss its types. The Exponential Equation Formula can be of three different forms. These are what they are:

  1. The same bases are used on both sides of the equations. For instance, 4x = 42
  2. Equations that can be converted to the same form using different bases. (For instance, 4x= 16 can be represented as 4x= 42)
  3. Equations that cannot be made to have the same base. (For instance, 4x =15)

Equations with Exponents

An Exponential Equation Formula is a function where each time x increases by an interval, the y-values are multiplied by the same quantity (the growth factor). Growth and decay are the two primary forms of exponential functions. When x increases, exponential growth functions grow while exponential decay functions shrink. The product of the exponential function ax with the natural log of a yields the derivative of the Exponential Equation Formula f(x) = ax, where a > 0. The following is a mathematical illustration of this using the integration of exponential functions:

f'(x)=ax ln a

When we plot a graph of an exponential function’s derivatives, the orientation of the graph changes when a > 1 and when a< 1.

Using the aforementioned technique, students can now determine the derivative of the exponential function ex. where e is also known as Euler’s number, a natural number. It is a crucial mathematical constant, and its value is 2.71828. (approx).

The study of exponential function characteristics is aided by an exponential function graph. The graph of x’s exponents demonstrates how the curve steepens as the exponent rises. The rate of expansion accelerates as well. An exponential function’s graph consists of an ascending or descending curve with a horizontal asymptote. A comprehensive understanding of the characteristics of exponential functions can be obtained from the graph of the fundamental exponential function y=ax shown below.

The graph strictly grows as x when a>1. Regardless of the value of a the graph will pass through (0,1) because a0 = 1. This graph demonstrates that it is entirely above the x-axis. The reason for this is that y’s range only includes positive real numbers.

Consider exponential growth, which is the idea of gradual rises followed by ones that increase fast, as one approach to conceptualise exponential functions. As a function of the variable x, these increases (or drops when using negative exponents) are constant over a predetermined length of time. The increases, for instance, are usually double or triple.

The Exponential Equation Formula can also be used in the business sense. Students can think about these two businesses:

Company A has 100 stores and adds 50 additional stores a year to grow.

Company B has 100 stores and grows by adding 50% more stores annually to its total.

Business A is expanding linearly. When there is linear growth, the rate of change is constant, which means that for every increase in input, the output increases by a fixed amount. The annual growth rate of new stores is constant for company A.

Company B is distinctive in that rate of change—rather than being defined as a fixed number of stores annually—is expressed as a percentage. To see the significance of this disparity, compare a 50% increase when there are 100 stores to a 50% increase when there are 1000 stores:

50 stores would be added that year if there were 100 stores and a 50% growth.

500 stores would be an increase of 50% from the 1000 stores in that year.

Many significant systems exhibit exponential development and decline. For instance, ambient radiation in the atmosphere after a nuclear catastrophe typically drops exponentially, but the number of germs in a colony often increases exponentially. Scientists can more accurately forecast outcomes by using data to plot a curve.

Exponential Equations Formula

The linear equation’s graph is a straight line with an upward trend that is constant or steady. Contrarily, the exponential equation’s graph shows an increase at a rising rate that forms a curve rather than a straight line. Since the powers in an Exponential Equation Formula, such as 23 = 2 * 2 * 2, specify how many times the base number is to multiply itself, students are continuously multiplying by the same value as the power rises. This relationship makes the graph appear curved or concave because y rises gradually initially and then quickly as the x values rise.

Some rules related to the Exponential Equations Formula are shown below-

  1. When the base is the same, the exponents are added after the bases are multiplied.
  2. The exponents are deducted from the division of the bases when the base is the same number.
  3. The base will remain the same and the exponents will multiply when a power has an exponent.
  4. The bases multiply and have the same power when two separate bases have the same exponents as power. Both the denominator and the numerator will have the same power or exponent when a fraction is raised to a power.

Property of Equality for Exponential Equations

An increase in quantity over time that starts off extremely slowly and subsequently picks up speed is referred to as exponential growth. Thus, as time passes, the rate of change increases. Exponential growth describes quick expansion. The exponential increase in population over time is depicted by the adjacent exponential growth curve. Exponential growth is defined by the following Exponential Equation Formula:

y = a ( 1+ r )x

Here r is the growth percentage.

The opposite of exponential growth is exponential decay. To examine bacterial infections, exponential growth and decay are frequently used. Exponential decay describes a quantity reduction over time that starts off quickly and then gradually slows down. As a result, change happens more slowly over time. An “exponential decrease” describes the quick fall. The exponential decay is defined by the following formula:

y = a ( 1- r )x

where r represents the rate of deterioration.

Exponential Equations to Logarithmic Form

Typically, one can let the independent variable to be the e in order to create an exponential function ( known as the exponent). The function utilising the exponential function graph is a straightforward example. Formula f(x)=2x (image will be uploaded soon) The Exponential Equation Formula increases rapidly, as shown in the exponential function graph of f(x) shown. For example, an exponential function appears in various straightforward models of bacterial growth. Exponential Equation Formula are solutions to the most basic types of dynamic systems. Decay or growth can be simply described by an exponential function. An illustration of exponential decay is provided by the function below.

g(x) = 1x/2

The function machine metaphor, which accepts inputs for variable x and changes them into outputs f, can be used to describe the behaviour of a given Exponential Equation Formula, f(x) (x). It is helpful to introduce parameters into a function using the function machine paradigm. The two exponential functions f(x) and g(x) above are distinct from one another, but their sole difference is the change from 2 to 1/2 in the exponentiation base. The Exponential Equation Formula is applicable to a variety of situations, including population increase, radioactive decay, and compound interest (money). However, the function is not exactly of the type f(x) = bx in the majority of instances. Constants are frequently added to or multiplied to make adjustments.

An illustration of exponential decay is radioactive decay. The half-life of radioactive elements is known. This is how long it takes for an element’s mass to decay in half and turn into something else. For instance, uranium-238 has a half-life of approximately 4.47 billion years and is a radioactive element that slowly decays. The conversion of 100 grammes of uranium-238 into 50 grammes of uranium-238 will take that long, according to this (the other 50 grammes will have turned into another element). Radon-220, on the other hand, has a half-life of roughly 56 seconds. In less than a minute, 100 grammes of radon-220 will split into 50 grammes of radon-220 and 50 grammes of another substance.

Solving Exponential Equations With Same Bases

It is necessary for students to practice questions related to Exponential Equation Formula with the same bases. Students having problems in solving them can take assistance from the Extramarks learning portal.

Solving Exponential Equations With Different Bases

All the questions that are asked regarding the Exponential Equation Formula with different bases can be practised well by taking help from Extramarks. Students need to practice questions from each chapter to score well in the Mathematics examination. Students need to revise the Exponential Equation Formula on a regular basis in order to solve questions in an appropriate manner. The NCERT solutions provided by Extramarks are very significant for practising questions related to the Exponential Equation Formula. The NCERT solutions consist of exact solutions to problems related to the Exponential Equation Formula.

Exponential Equations Examples

The Exponential Equation Formula needs to be learned by students. The Exponential Equation Formula is crucial for practising examples. All the questions can be solved if the Exponential Equation Formula is applied properly.

Exponential Equations Questions

Students are supposed to keep revising the Exponential Equation Formula to solve questions. The Exponential Equation Formula is an important topic that students need to master.

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FAQs (Frequently Asked Questions)

1. What is the Exponential Equation Formula?

The Exponential Equation Formula is given as f(x) = a(x). This can be easily learned by practising questions from time to time. Numerous questions that involve the Exponential Equation Formula are provided on the Extramarks educational website.

2. Where can students find the solutions to problems related to the Exponential Equations Formula?

Extramarks has solutions related to the Exponential Equation Formula. Students can easily download NCERT solutions that can be used to solve questions specific to the Exponential Equation Formula.