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Exponential Equations

The exponential equation is in the form of y = ax, where 'a' is positive real number and x is the variable. If base of the exponential equation is same on the equal sides, then the equation solved by using the property, if bx = by, then x = y, where b > 0 and b $\neq$ 0.

For example, the equations x2 - 2 = 0 and x4 - 4 x2 + 4 = 0 are exponential equations.

Below are given examples of solving exponential equations with steps of explanation:

Exponential Equation 1:

Solve 32x–1 = 27

Solution:           

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

Solve the equation and 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

Exponential Equation 2:

Solve for x in the equation ex = 80

Solution:

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

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

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

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

Example: Solve for x in the equation 10x + 5 - 8 = 60

Solution:

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

Step 2: Take the common log of both sides:

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

Step 3: Simplify the left part of the above equation using Logarithmic Rule 3:

                               (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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