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# Exponential Equation

An equation in which the variable appear in the exponent, called an exponential equation. The exponential equation is in the form of y = ax , Where "a" is positive real number and "x" is the real number is known as the exponential equation. Population growth, inflation, and radioactive decay are few examples of the various phenomenon that exponential functions can be used to model.

 Related Calculators Exponential Equation Calculator Calculate Exponential Exponential Distribution Calculator Graphing Exponential Functions Calculator

## Exponential Equation Definition

An exponential equation is one in which a variable occurs in the exponent (may be positive or negative). An exponential equation in which each side can be expressed in terms of the same base can be solved by using the property, if bx = by, then x = y, where b ≠ 0.

If the bases of exponential equation are same, then exponents are equal.

## Solving Exponential Equations

To solve an exponential equation, follow the steps given below:

• First isolate the exponential function by rewriting the equation with a base raised to a power on one side.
• Take the logarithmic each side of the function.
• Solve for the variable using logarithm laws and properties.

### Solving Exponential Equations with Logarithms

Logarithm laws and properties can be used to solve exponential equations. To express each side of an exponential equation as the power of the same base, take the logarithm of each side of the equation.

If $4^x$ = 28, then log($4^x$) = log 28

$x$ log 4 = log 28 (Using logarithmic property: log($a^b$) = b log (a)

x = $\frac{log\ 28}{log\ 4}$ $\approx$ 2.40

### Solving Exponential Equations with Different Bases

Let us Solve 32x - 1 = 27

In this case, exponential on one side of the "equals" and a number on the other.

Solve the equation the express the "27" as a power of 3. Since 27 = 33,

32x - 1 = 27

32x - 1 = 33

2x - 1 = 3

2x = 4

x = 2

The Exponential equation answer x is 2

### Solving Exponential Equations with e

Solve for x in the equation ex = 80

Step 1: Take the natural log of both sides: Ln(ex) = Ln(80)

Step 2: Simplify the left part of the above equation using Logarithmic Rule ---> x Ln(e) = Ln(80)

Step 3: Simplify the left part of the above equation: Since Ln (e) =1, the equation reads x = Ln(80)

Ln(80) is the exact answer and x = 4.38202663467 is an approximate answer because we have rounded the value of Ln(80).

### Solving Exponential Equations with Fractions

Solve for x,

$(\frac{1}{2})^{5x - 2}$ = $\frac{1}{32}$

$(\frac{1}{2})^{5x - 2}$ = $\frac{1}{32}$

$(\frac{1}{2})^{5x - 2}$ = $(\frac{1}{2})^5$

The base of the exponential equation is the same on the equal sides, then the equation solved by using the property, if bx = by, then x = y, where b > 0 and b ≠ 0.

5x - 2 = 5

5x = 7

x = $\frac{7}{5}$.

## Simplifying Exponential Equations

While there is no particular formula for solving an exponential equation. Logarithm laws and properties can be used to solve exponential equations. There are common techniques used in finding the unknown value in an exponential equation.

### Rearranging Exponential Equations

Solve for x, 3x2 - 4 = 71

Isolate x2 by adding 4 to both sides then dividing both side by 3.

3x2 - 4 + 4 = 75

3x2 = 75

x2 = 25

To isolate x, raise to the inverse exponent, ie. $\frac{1}{2}$

$(x^2)^{\frac{1}{2}}$ = $(25)^{\frac{1}{2}}$

x = $(5^2)^{\frac{1}{2}}$

x = 5

### Factoring Exponential Equations

Using factoring rules, simplify and solve the exponential equation, x2 + 7x + 10 = 0.

x2 + 7x + 10 = x2 + 5x + 2x + 10

= x(x + 5) + 2(x + 5)

= (x + 2)(x + 5) = 0

=> x = -2, -5.

## Exponential Differential Equation

To find the derivatives of exponential functions, we must use some rules:

Exponential function with base e, is its own derivative.

$\frac{d}{dx}$ $e^x$ = $e^x$

The derivative of f(x) = ax is given by

f '(x) = ax ln a

Find the derivative of f(x) = 10x + x

f'(x) = 10x ln 10 + 1

= 10x + 1 (Because ln 10 = 1)

## Graphing Exponential Equations

Graphing exponential functions is similar to the graphing other functions. Consider the exponential function, f(x) = ax, for every x in R with a $\neq$ 1.

Graph of f for a > 1 and for 0 < a < 1

Let f(x) = an exponential function with a > 1.

Let g(x) = an exponential function with 0 < a < 1.

## Exponential Model Equation

An exponential model has an equation of the form, f(x) = a bx, where a $\neq$ 0 and b > 0. The percentage change is (b - 1) 100 % and the parameter "a" is the output corresponding to an input of zero. The graph of an exponential model with positive "a" is given below:

Exponential growth f(x) = a bx , with b > 1

Exponential decay f(x) = a bx , with 0 < b < 1

## Exponential Equation Examples

Given below are some of the examples on exponential equations.

### Solved Example

Question: Solve for x in the equation $10^{x + 5}$ - 8 = 60
Solution:

Step 1: Isolate the exponential term before the take the general log of both sides. Therefore, add 8 to both sides: $10^{x + 5}$ = 68

Step 2: Take the common log of both sides:

Log ($10^{x + 5}$) = Log(68)

Step 3: Simplify the left part of the above equation using Logarithmic Rule, log $a^x$ = x log a:

(x + 5)Log (10) = Log (68)

Step 4: Simplify the left part of the above equation: Since Log (10) = 1, the above equation are given (x + 5) = log (68)

Step 5: Subtract 5 from both sides of the above equation:

x = Log (68) - 5

x = -3.16749108729 is an approximate answer.

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