Solving for a missing exponent can be as simple as solving 4=2^x, or as complex as finding how much time must pass before an investment is doubled in value. (Note that the caret refers to exponentiation.) In the first example, the strategy is to rewrite the equation so both sides have the same base. The latter example may take the form principal(1.03)^years for the amount in an account after earning 3 percent annually for a certain number of years. Then the equation to determine the time to doubling is principal(1.03)^years=2*principal, or (1.03)^years=2. One then needs to solve for the exponent "years (Note that asterisks denote multiplication.)
Things You'll Need
- Calculator that performs logarithms (optional)
Move the coefficients over to one side of the equation.
For example, suppose you need to solve 350,000=3.5*10^x. Then divide both sides by 3.5 to get 100,000=10^x.
Rewrite each side of the equation so the bases match.
Continuing with the example above, both sides can be written with a base of 10. 10^6 = 10^x.
A harder example is 25^2=5^x. The 25 can be rewritten as 5^2. Note that (5^2)^2=5^(2*2)=5^4.
Equate the exponents. For example, 10^6=10^x means x must be 6.
Take the logarithm of both sides instead of making the bases match. Otherwise, you may have to use a complex logarithm formula to make the bases match.
For example, 3=4^(x+2) would need to be changed into 4^(log 3/log 4)=4^(x+2). The general formula for making bases equal is: base2=base1^(log base2 / log base1).
Or you could just take the log of both sides: ln 3=ln [4^(x+2)]. The base of the logarithm function you use does not matter. The natural log (ln) and the base-10 log are equally fine, as long as your calculator can calculate the one you pick.
Bring the exponents down in front of the logarithms.
The property being used here is log (a^b)=blog a. This property can intuitively be seen to be true if you now that log ab=log a + log b. This is because, for example, log (2^5)=log(22222)=log2+log2+log2+log2+log2=5log2.
So for the doubling problem stated in the introduction, log (1.03)^years=log 2 becomes yearslog (1.03)=log 2.
Solve for the unknown like any algebraic equation. Years=log 2 / log(1.03).
So to double an account paying an annual rate of 3 percent, one must wait 23.45 years.
How to Graph Exponential Functions, an Easy Way
The Graphs of Exponential functions can be easily sketched by using three points on the X-Axis and three points on the Y-Axis....
How to Find the Missing Number in an Equation
In algebra, finding the missing number in an equation is known as “solving for x.” Because we don’t know what the missing...
How to Find the Base of a Number
To find a number's base, it helps to have some background knowledge of powers. Powers are composed of bases and exponents. For...
How to Find Exponents That Are Variables
The most surefire way to find exponents that are variables is to use logarithms. Other algebraic methods exist, but they only work...
How to Solve Exponents With Different Bases
In an exponential function, the number that you will be raising to a power is the base and the power is the...
How to Determine an Unknown Exponent
To solve an equation for the exponent, use natural logs in order to solve the equation. Sometimes, you can perform the calculation...