You will learn how to solve exponential equations by taking the common logarithm of each side of an exponential equation.
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
Sep 01, 18 04:07 PM
These heart of algebra questions will help you prepare to take the math portion of the SAT
New math lessons
Your email is safe with us. We will only use it to inform you about new math lessons.
Sep 01, 18 04:07 PM
These heart of algebra questions will help you prepare to take the math portion of the SAT
Our Top Pages
Formula for percentage
Compatible numbers
Basic math test
Basic math formulas
Types of angles
Math problem solver
Algebra word problems
Surface area of a cube
Finding the average
Scale drawings
Everything you need to prepare for an important exam!
K-12 tests, GED math test, basic math tests, geometry tests, algebra tests.
Real Life Math Skills
Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball.