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Solving Exponential Equations

When the variable we want to solve for is in the exponent, we again "do the same thing" to both sides of the equation. This time we take logarithms, using the property that
a=b => log(a)=log(b) (or ln(a)=ln(b))
 The logarithm rule
log(m^x) = x log(m)
 is also useful.

Examples
For each, we solve for x. Notice how we use the log(x) and ln(x) functions to eliminate base 10 and base e exponentials, respectively, and how we use the logarithm rule for other bases:
  1. 10^(2x+1) = 3, so log(10^(2x+1)) = log(3), or 2x+1 = log(3), and x = (log(3) - 1)/2
    Note that this solution is exact. The decimal approximation x approx -0.2614 gives an approximate solution.
  2. 2e^(2x) = 12, so e^(2x) = 6, or ln(e^(2x)) = ln(6), so 2x = ln(6), or x = ln(6)/2
    Note that we divided both sides by two before taking the log to make life easier for ourselves.
  3. 1.07 = 4^(-x), so ln(1.07) = ln(4^(-x)), or ln(1.07) = -x ln(4), and x = -ln(1.07)/ln(4)
    Note that we could solve this with the base 10 log as well:
    1.07 = 4^(-x), so log(1.07) = log(4^(-x)), or log(1.07) = -x log(4), and x = -log(1.07)/log(4)
    Check with your calculator that these two answers are the same!
  4. P = P0 e^(rx), or (P/P0) = e^(rx), so ln(P/P0) = ln(e^(rx)), or ln(P/P0) = rx, so x = (1/4)ln(P/P0)
Practice

Be sure you've gone through each step in the examples above before doing these. Once you've worked them until you're sure that you understand them, go on to the next section. There are more problems of this type in the section test at the end of the section.

Note that you can get new practice problems by clicking the "Refresh" button at the bottom of the practice set.

  1. Solve for x:
    a e ^b ^x ^+ ^c   =   d
    Answer (give an approximate answer to at least 3 decimal places):   x =
    (Enter answers, then click: . Answer message: )
  2. And then solve this equation for x:
    g ( 10 ^h ^x )   =   j ( 10 ^k ^x )
    Answer (give an approximate answer to at least 3 decimal places):   x =
    (Enter answers, then click: . Answer message: )
For more practice, click: .
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precal: 7.5 - solving exp eqns
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©2003-2007 Gavin LaRose, Pat Shure / University of Michigan Math Dept. / Regents of the University of Michigan