An equation in which the variable appear in the exponent, called an exponential equation. The exponential equation is in the form of y = a^{x} , 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.

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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 b^{x} = b^{y}, then x = y, where b ≠ 0.

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

### Solving Exponential Equations with Logarithms

### Solving Exponential Equations with Different Bases

### Solving Exponential Equations with e

### Solving Exponential Equations with Fractions

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, 3x^{2} - 4 = 71

Isolate x^{2} by adding 4 to both sides then dividing both side by 3.

3x^{2} - 4 + 4 = 75

3x^{2} = 75

x^{2} = 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, x^{2} + 7x + 10 = 0.

x^{2} + 7x + 10 = x^{2} + 5x + 2x + 10

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

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

=> x = -2, -5.

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) = a^{x} is given by

f '(x) = a^{x} ln a

Find the derivative of f(x) = 10^{x} + x

f'(x) = 10^{x} ln 10 + 1

= 10^{x} + 1 (Because ln 10 = 1)

Graphing exponential functions is similar to the graphing other functions. Consider the exponential function, f(x) = a^{x}, 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.

An exponential model has an equation of the form, f(x) = a b^{x}^{}, 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 b^{x} , with b > 1

Exponential decay f(x) = a b^{x} , with 0 < b < 1

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: **

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

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.

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

Let us Solve 3^{2x - 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 = 3^{3},

3^{2x - 1} = 27

3^{2x - 1} = 3^{3}

2x - 1 = 3

2x = 4

x = 2

The Exponential equation answer x is 2

Solve for x in the equation e^{x }= 80

**Step 1:** Take the natural log of both sides: Ln(e^{x}) = 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).

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 b^{x} = b^{y}, then x = y, where b > 0 and b ≠ 0.

5x - 2 = 5

5x = 7

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

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.

Solve for x, 3x

Isolate x

3x

3x

x

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

Using factoring rules, simplify and solve the exponential equation, x

x

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

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

=> x = -2, -5.

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) = a

f '(x) = a

Find the derivative of f(x) = 10

f'(x) = 10

= 10

Graphing exponential functions is similar to the graphing other functions. Consider the exponential function, f(x) = a

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.

An exponential model has an equation of the form, f(x) = a b

Exponential growth f(x) = a b

Exponential decay f(x) = a b

Given below are some of the examples on exponential equations.

**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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