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
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
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
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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