You will learn how to solve exponential equations by taking the common logarithm of each side of an exponential equation. Generally speaking, follow these four steps when solving exponential equations.
Example #1
2^{3x} = 64
Take the common logarithm of each side.
log 2^{3x} = log 64
Use the power property of logarithms
3x log 2 = log 64
Divide each side by 3 log 2 to isolate x
x = log 64 / 3 log 2
x = 1.8 / 3 x 0.3 ( log 64 and log 2 were rounded to the nearest 10th)
x = 1.8 / 0.9
x = 2
Notice that you could have solved the problem above this way too
2^{3x} = 64
2^{3x} = 2^{6}
3x = 6
x = 2
However, this works only if you can easily rewrite the number on the right or 64 so that it will have the same base or 2 as the expression on the left. For the next example, it is not possible to easily rewrite the expression so that the same base shows up. Can you tell what y is for this expression 2^{y} = 60 ?
Therefore, it is easier to take the log of both sides.
Example #2
2^{3x} = 60
Take the common logarithm of each side.
log 2^{3x} = log 60
Use the power property of logarithms
3x log 2 = log 60
Divide each side by 3 log 2 to isolate x
x = log 60 / 3 log 2
x = 1.78 / 3 x 0.3 ( log 60 and log 2 were rounded to the nearest 10th)
x = 1.78 / 0.9
x = 1.97
Example #3
5^{2x+4} = 120
Take the common logarithm of each side.
log 5^{2x+4} = log 120
Use the power property of logarithms
(2x + 4) log 5 = log 120
2x log 5 + 4 log 5 = log 120
2x log 5 = log 120 - 4 log 5
Divide each side by 2 log 5 to isolate x
x = ( log 120 - 4 log 5 ) / 2 log 5
x = ( 2.079 - 4 x 0.699 ) / 2 x 0.699 ( log 120 and log 5 were rounded to the nearest thousandth)
x = ( 2.079 - 2.796 ) / 1.398
x = -0.717 / 1.398
x = -0.512
Example #4
Solve 3^{4x + 3} - 8^{-x+2} = 0
Step 1: Isolate the exponential expression
3^{4x + 3} = 8^{-x+2}
Step 2: Take the common logarithm of each side
log 3^{4x + 3 }= log 8^{-x+2}
Step 3: Use the power property of logarithms
(4x + 3) log 3 = (-x + 2) log 8
Step 4: Solve for the variable
4x log 3 + 3 log 3 = -x log 8 + 2 log 8
4x log 3 + x log 8 = 2 log 8 - 3 log 3
x(4 log 3 + log 8) = 2 log 8 - 3 log 3
x = (2 log 8 - 3 log 3) / (4 log 3 + log 8)
x = (2 × 0.90 - 3 × 0.47) / (4 × 0.47 + 0.90)
x = 0.39 / 2.79 = 0.139
May 07, 21 02:29 PM
A time-series data shows information about the same subject or element of a sample or population for different periods of time.
Basic math formulas
Algebra word problems