Solve Exponential Functions
We must separate the exponentiating variable to solve exponential functions. With “a” as the base and “x” as the exponent, exponential functions have the general form f(x) = axe. The objective is to identify the value of “x” that resolves the inequality or equation. There are various options depending on the precise issue and the desired resolution.
We can frequently utilize logarithms to remove the variable from the exponent while solving exponential equations. We can simplify the equation and immediately solve for the variable by taking the logarithm of both sides of the equation. The exact challenge at hand, as well as any existing constraints, will determine the logarithm basis to use.
Let’s use the equation 2 x 16 as an example. By taking the logarithm of both sides and using a logarithm base to solve for “x,” we may make the problem simpler. Since the exponential function’s base is two, we can utilize the logarithm base 2 in this situation. Logarithmically speaking, log2(2x) = log2(16).
We reduce the equation to x = log2(16) using the logarithm property that says loga(ab) = b. x = 4 is the result of evaluating the right-hand side of the logarithm. Consequently, x = 4 is the equation 2x = 16’s answer.
The strategy is similar when dealing with inequalities using exponential functions. We can utilize logarithms to remove the variable from the exponent and directly answer the inequality. It’s crucial to remember that only when the base of the logarithm is positive do logarithms maintain the inequality’s order.
It is essential to look for any constraints on the function’s domain or any potential superfluous solutions that might emerge throughout the solving process while solving exponential functions. Furthermore, the solutions could not be limited to real numbers if the function involved complex numbers.
What Are Exponential Equations?
An exponential equation is an equation with exponents where the exponent (or a part of the exponent) is a variable. For example, 3x = 81, 5x – 3 = 625, 62y – 7 = 121, etc. are examples of exponential equations. We may come across the use of exponential equations when solving the problems of algebra, compound interest, exponential growth, exponential decay, etc.
Types of Exponential Equations
There are three types of exponential equations. They are as follows:
- Equations with the same bases on both sides. (Example: 4x = 42)
- Equations with different bases can be made the same. (Example: 4x = 16 which can be written as 4x = 42)
- Equations with different bases cannot be made the same. (Example: 4x = 15)
Equations With Exponents
We will go into the subject of equations with exponents, investigating their characteristics, norms, and solution methods.
Exponents are a quick way to represent repeated multiplication. They are also referred to as powers or indexes. The number of times a base has been multiplied by itself is represented by an exponent. For instance, the base and exponent in the expression 23 are 2 and 3, respectively. It denotes a three-fold multiplication of 2 by itself:
2^3 = 2 × 2 × 2 = 8
Calculations are made easier by the characteristics of exponents. One important characteristic is the product of powers, which causes the exponents to be added when two integers with the same base but different exponents are multiplied.
a(m+ n) = a(m a)
The power of a power is another characteristic, according to which increasing one exponent by another exponent multiplies the exponents:
(a m ) n = a m n
An understanding of these properties is necessary for manipulating and resolving equations with exponents.
Exponentiation Of Equations
We must use techniques that decompose the equation and isolate the variable to solve equations with exponents. Here are several methods that are frequently used:
- Simplify both sides: Begin by separating the two sides of the equation. Expressions can be made simpler by combining like terms and using exponent rules.
- Remove the exponent: Remove the exponent using inverse procedures. Take the square root of both sides to remove the exponent, like when the variable is increased to the power of 2.
- Applying logarithms: Sometimes help resolve equations involving exponents. Since logarithms are exponentiation’s inverse operations, they can isolate the variable. Use logarithms with the same base as the exponent on both sides of the equation.
- Factorization: Remove the common base from the equation if any terms have the same base but distinct exponents. You can then solve for the variable and simplify the equation.
- Substitution: In some cases, replacing the exponentiated variable with a different one can simplify the equation. This method introduces a new variable to represent the original variable multiplied by a specific power.
You can solve for the variable and locate the solution to equations with exponents by using these methods and carefully altering the equation.
Exponential Equations Formulas
While solving an exponential equation, the bases on both sides may be the same or not. Here are the formulas used in each case, which we will learn in detail in the upcoming sections.
Property Of Equality For Exponential Equations
This property is useful for solving an exponential equation with the same bases. It says that when the bases on both sides of an exponential equation are equal, the exponents must also be equal. i.e.,
ax = ay, x = y.
Exponential Equations To Logarithmic Form
We know that logarithms are nothing but exponents and vice versa. Hence, an exponential equation can be converted into a logarithmic function. This helps in the process of solving an exponential equation with different bases. Here is the formula to convert exponential equations into logarithmic equations.
bx = a ⇔ log a = x
Solving Exponential Equations With Same Bases
Sometimes, an exponential equation may have the same base on both sides. For example, 5 x 53 has the same base five on both sides. Sometimes, though the exponents on both sides are not the same, they can be made the same.
For example, 5x = 125. Though it doesn’t have the same bases on both sides of the equation, they can be made the same by writing it as 5x = 53 (as 125 = 53). To solve the exponential equations in each of these cases, we just apply the property of equality to the exponential equations. We set the exponents to be the same for the variable.
Here is another example where the bases are not the same but can be made to be the same.
Example: Solve The Exponential Equation 7y + 1 = 343y
We know that 343 is 73. Using this, the given equation can be written as,
7y + 1 = (73)y
7y + 1 = 73y
Now the bases on both sides are the same. So we can set the exponents to be the same.
y + 1 = 3y
Subtracting y from both sides,
2y = 1
Dividing both sides by 2,
y = ½
Solve the equation 3x = 81
In this example, we have the base three raised to the power of the variable x, and the equation is set equal to 81. Our goal is to determine the value of x that satisfies the equation.
To solve this equation, we can use the property of logarithms that states that if axax = b, the log(b(b)) = x. Applying this property, we can take the logarithm of both sides of the equation with the base 3:
x = log₃(81)
Now, we need to evaluate the logarithm of 81 with base 3. The logarithm represents the exponent to which the base must be raised to obtain the argument. In this case, we want to find the exponent to which three must be raised to obtain 81. Evaluating this logarithm yields:
x = log₃(81) = 4
Therefore, the solution to the equation 3x = 81 is x = 4. By substituting x = 4 back into the original equation, we can verify that 34 equals 81.
Solving Exponential Equations With Different Bases
Sometimes, the bases on both sides of an exponential equation may not be the same. We solve the exponential equations using logarithms when the bases differ. For example, 5x = 3, which neither has the same bases on both sides nor can the bases be made the same. In such cases, we can do one of the following things:
- Convert the exponential equation into the logarithmic form using the formula bx = a ⇔ log a = x and solve for the variable.
- Apply logarithm (log) on both sides of the equation and solve for the variable. In this case, we must use a logarithm property, log am = m log a.
We will solve the equation 5x = 3 in each of these methods.
We will convert 5 x 3 into logarithmic form. Then we get,
log53 = x
Using the change of base property,
x = (log 3) / (log 5)
We will apply the log on both sides of 5x = 3. Then we get log 5 x log 3. Using the property log am = m log an on the left side of the equation, we get x log 5 = log 3. Dividing both sides by log 5,
x = (log 3) / (log 5)
Important Notes On Exponential Equations
Here are some important notes concerning the exponential equations:
- To solve the exponential equations of the same bases, just set the exponents equal.
- Applying logarithms on both sides to solve the exponential equations of different bases.
- The exponential equations with the same bases also can be solved using logarithms.
- If an exponential equation has one on any one side, then we can write it as 1 = a0 for any ‘a. For example, to solve 5x = 1, we can write it as 5x = 50, then we get x = 0.
- To solve an exponential equation using logarithms, we can apply “log” or “ln” on both sides.
What is an exponential function?
An exponential function is a mathematical function in which the variable appears as an exponent. It has the form f(x) = a * b^x, where “a” and “b” are constants. The base “b” is usually greater than 1, and as “x” increases, the function grows or decays exponentially.
How do you solve an exponential equation?
To solve an exponential equation, you typically want to isolate the exponential term. Take the natural logarithm (ln) of both sides of the equation to remove the exponential. Then, solve for the variable using algebraic techniques. Remember to check for extraneous solutions if you’re raising both sides to a power.
What is the exponential growth formula?
The exponential growth formula is given by the equation P(t) = P₀ * e^(rt), where P(t) represents the final value after time “t,” P₀ is the initial value, “e” is the base of the natural logarithm (approximately 2.71828), and “r” is the growth rate.
What is exponential decay?
Exponential decay refers to a process in which a quantity decreases over time according to an exponential function. The general form is given by f(x) = a * b^(-x), where “a” is the initial value, “b” is the base (usually between 0 and 1), and “x” represents time or the number of intervals.
How do you graph an exponential function?
To graph an exponential function, determine key points by substituting different values of “x” into the function. Plot these points on a coordinate plane and connect them with a smooth curve. Remember that the base of the exponential function determines the behavior of the graph: if it’s greater than 1, the graph will exhibit exponential growth, and if it’s between 0 and 1, it will show exponential decay.
What are some real-life applications of exponential functions?
Exponential functions are prevalent in various fields. Some examples include population growth, compound interest calculations, radioactive decay, the spread of diseases, and the charging/discharging of capacitors or batteries. Exponential functions are useful for modeling phenomena that experience rapid growth or decay over time.