Exponential Equations

Exponents are used in exponential equations, as the name implies. The exponent of a number (base) indicates how many times the number (base) has been multiplied. But what happens when a number’s power is a variable? An exponential equation is one in which the power is a variable and is a part of an equation. To answer exponential equations, we may need to apply the relationship between exponents and logarithms. 

Exponential equations: 

An exponential equation is one in which the exponent (or a portion of the exponent) is a variable. Examples of exponential equations include 3^x = 81, 5^{x – 3} = 625, 6 ^{2y – 7} = 121, and so on. When tackling problems involving algebra, compound interest, exponential growth, and exponential decay, we may encounter the application of exponential equations. 

Types of exponential equations: 

Exponential equations are divided into three categories. These are the details:

• Equations on both sides with the same foundation. (For instance, 4x = 42)
• It is possible to make equations with different bases the same. (For instance, 4x = 16 can be represented as 4x = 42.)
• Equations with various bases that can’t be combined. (For instance, 4x = 15) 

Exponential equations formula: 

The bases on both sides of an exponential equation may or may not be the same when solving it. The formulas utilised in each of these circumstances are listed below, and we’ll go through them in-depth in the next sections. 

• When bases are saame, apply 

ax = ay ←→ x = y 

• When bases are not same, apply 

bx = a ←→ logb a = c 

Property of Equality for Exponential Equations: 

When solving an exponential equation with the same bases, this characteristic comes in handy. When the bases on both sides of an exponential equation are equal, the exponents must likewise be equal, according to this rule. i.e., 

ax = ay ⇔ x = y. 

Exponential Equations to Logarithmic Form: 

We all know that exponents are logarithms and vice versa. As a result, a logarithmic function can be created from an exponential equation. This aids in the solution of an exponential equation with multiple bases. The formula for converting exponential equations to logarithmic equations is given below. 

bx = a ⇔ logba = x 

Solving Exponential Equations With Same Bases: 

On both sides of an exponential equation, the bases might sometimes be the same. 5x = 53, for example, has the same base 5 on both sides. Even though the exponents on both sides are not the same, they can sometimes be made to be the same. For instance, 5x Equals 125. Even if the bases on both sides of the equation are not the same, they can be written as 5x = 5³ (as 125 = 5³). In each of these circumstances, we simply apply the property of equality of exponential equations, which allows us to set the exponents to the same value and solve for the variable.

Solving Exponential Equations With Different Bases: 

The bases on both sides of an exponential equation aren’t always the same (and can’t always be made the same). When the bases on both sides of the equation are not the same, we solve exponential equations with logarithms. 5x = 3, for example, does not have the same bases on both sides and cannot be made the same. In such instances, one of the following options is available. 

• Solve the variable by converting the exponential equation to logarithmic form using the method bx = a logba = x.
• Solve for the variable using the logarithm (log) on both sides of the equation. In this scenario, we’ll have to employ the logarithm property log am = m log a. 

Solving exponential equations algebraically: 

You’ll need some mathematics to solve exponential equations. If the exponential equation is more involved, you may need to isolate the base using algebraic techniques. To solve for the missing variable, apply the property of equality or logarithms once the base has been isolated. 

Points to remember: 

Here are some key points to remember about exponential equations.

• Set the exponents equal to solve the exponential equations with the same bases.
• Apply logarithm on both sides to solve exponential equations with distinct bases.
• Logarithms can also be used to solve exponential equations with the same bases.
• If one of the sides of an exponential equation is 1, we can write it as 1 = a0 for any ‘a’. To solve 5x = 1, we can write it as 5x = 50, which gives us x = 0.
• We can use either “log” or “ln” on both sides to solve an exponential equation using logarithms. 

Solved examples: 

Q1. Solve 27 / (3-x) = 36.

Solution:

We know that 27 = 3³. We can make the bases to be the same on both sides using this.

3³/ (3-x) = 36

Using the quotient property of exponents, am/an = am – n. Using this,

3³ – (-x) = 36

3³+ x = 36

Now the bases on both sides are the same. So we can set the exponents to be equal.

3 + x = 6

Subtracting 3 from both sides,

x = 3.

Therefore, the solution of the given exponential equation is x = 3. 

Conclusion: 

Exponential equations are essential in research because they may be used to calculate growth, decay, time, and the amount of anything at a particular time. This session covers the history of exponential equations as well as how to graph them.