Exponential Equations

Exponential equations are those where the variable is used as an exponent. In an exponential equation, the variable appears in the exponent, instead of multiplying or adding many times. These equations show how something grows or shrinks quickly, like population growth or compound interest. Solving them often means figuring out what power you have to raise a number to get another number.
 

Exponential equations come in various types, including those with the same base, different bases, and those involving logarithms. Understanding each type helps in choosing the right method, like matching bases or applying logarithms, to simplify and solve the equation.
There are two main types of exponential equations.

Same Base Exponential Equations

These equations have the same base on both sides of the equation. Since the bases match, you can set the exponents equal and solve.
Example: 3x+2 = 35    
x+2 = 5
x = 5-2
x = 3
so, 33+2 = 35
 

When we have equations with different bases (like 2 and 3), they cannot be written in the same form.
2x = 5

Equations with Different Bases That Can Be Made the Same

When solving exponential equations where the bases on both sides can be made the same, logarithms are the most effective method.
8x = 64
Expressing both sides with the same base, we get
8 = 23
64 = 26
Now, we will rewrite the equation, (23)x = 26
23x = 26
3x = 6  x = 2

Exponential equation formulas help solve problems where the variable is in the exponent. These formulas include the basic exponential form, equality property, logarithmic conversion, and growth or decay models, often used in science, finance, and real-life change situations.

1. Basic Exponential Form

Basic Exponential Form is a way to express repeated multiplication of the same number using a base and an exponent. For example, 2³ means 2 × 2 × 2, which equals 8.
y = ax
Where:
a is the base (positive, not 1)

x is the exponent (can be a variable)

y is the result
 

2. Property of Equality for Exponents

In this property, if two exponential expressions with the same base are equal, then their exponents must also be equal.
If ax = ay, then x = y

3. Using Logarithms to Solve Exponentials

To solve equations where the variable is in the exponent, we can use logarithms. This helps us find the unknown exponent more easily.
If you can't match the bases:
ax = b  x = loga b
Or using common logarithms (base 10) or natural logs (base e):
x = log blog a or x = In bIn a

4. Exponential Growth and Decay Formulas

Exponential growth and decay describe how things can increase or decrease quickly over time. We follow special formulas to show this change.
Growth: y = a(1+r)t
Decay: y = a(1-r)t

Where:
a  Is the initial value

r Is the rate (as a decimal)

t is time

y Is the final amount
 

The Property of Equality for Exponential Equations states that if two exponential expressions have the same base, their exponents must be equal. This property helps solve equations by allowing you to set the exponents equal when the bases match.

The Property of Equality for Exponential Equations is a fundamental rule in solving exponential equations. It states that if two exponential expressions with the same base are equal, then their exponents must also be equal.
In mathematical terms:

ax = ay  x = y

This property applies when the bases a are the same and are positive numbers (except for 1). It's a powerful tool for solving equations where the variable is in the exponent.

For example, in the equation:
52x = 56

Since both sides have the same base (5), you can set the exponents equal:
2x = 6  x = 3

This simplifies the process of solving exponential equations efficiently.
 

An exponential equation is like a math sentence where a number is raised to a power. For example: 2x = 8. Here, 2 is the base, x is the exponent, and 8 is the result.
A logarithm is the opposite of an exponent. It helps us find the exponent when we know the base and the result. In the equation 2x = 8, the logarithmic form is x = log₂(8).
Identify the base, exponent, and result.

Write it as: exponent = log(base)(result).

For example:
32 = 9 becomes 2 = log₃(9)

General Conversion Rule:

If you have an exponential equation of the form:
ax = y 

It can be rewritten in logarithmic form as:
x = logay
Where:
a  is the base,

x  is the exponent (the unknown),

y is the result.
Example:
For an equation like 5x = 20, rewrite it as x = log520
This form allows you to solve x  using a calculator or logarithm rules. This conversion is especially helpful when the exponent cannot be easily solved by inspection or simple algebraic manipulation.
 

Exponential equations with the same bases are like puzzles where both sides have the same number raised to different powers. When the bases match, the exponents must also be equal for the equation to be true.

General Steps:

Match the bases: Ensure both sides of the equation have the same base.
Set exponents equal: If ax = ay, x = y.

Solve for the variable: After equating the exponents, solve the resulting algebraic equation.

Example 1:
32x = 34
Here, the bases are the same; set the exponents equal:
2x = 4  x = 2

Example 2:
5x+1 = 53
Again, the bases are the same (both 5), so set the exponents equal:
x+1 =  x = 2
 

We come across equations where the numbers with different bases are raised to variable exponents. For example, in the equation 2x = 5x-1, the numbers 2 and 5 are raised to powers, but they have different bases. This means they’re not the same number with different exponents, so we can’t rewrite them to look the same. We will follow these steps 

Convert the exponential equation into logarithmic form using the rule: If bx = a, then it's equivalent to log⁡ba = x.
Alternatively, you can take the logarithm of both sides of the equation and solve for the unknown.
In doing so, use the logarithmic identity:

log⁡ am = m log a, which allows you to bring the exponent in front as a multiplier. 

Method 1:

We start with:
4x = 9
Now we convert the exponential equation into logarithmic form using the formula: bx = a ⇔ logba = x
log⁡4 9=x

To change the base, we use the change of base formula:
 x=log9log 4

Method 2:

We start with:
2x = 7
Now we are taking the logarithm of both sides 
log⁡ 2x = log 7
Using the property  log am = m log a
x log 2 = 7
Use the change of base formula to change of base:
x=log 7log 2
 

Exponential equations have real-life applications in areas like population growth, radioactive decay, compound interest, and disease spread, where quantities change rapidly over time and follow exponential patterns.

• Population Growth: Exponential equations are used to model the growth of populations, The population increases by a constant percentage over time.

• Radioactive Decay: Radioactive substances decay at a predictable rate, with the amount of the substance reducing over time. This process is modeled by exponential decay equations, where the quantity of the substance decreases by a fixed percentage per unit of time.

• Compound Interest (Finance): Exponential equations are widely used in finance to calculate compound interest. When you invest money, the interest earned over time is added back to the principal, causing the investment to grow exponentially. This is commonly seen in savings accounts, loans, and investments.

• Viral Spread (Epidemiology): In the early stages of a virus outbreak, the number of infected individuals grows exponentially as each person spreads the disease to several others. This helps predict how quickly the disease might spread within a population.

• Sound and Light Intensity: The intensity of sound or light decreases exponentially as it moves away from the source. This principle is used in acoustics, where the loudness of sound decreases with distance, and in optics, where the brightness of light decreases as it travels through a medium.