Solving Exponential and Logarithmic Equations
To solve an exponential equation, the following property is sometimes helpful:
If a > 0, a ≠ 1, and ax = ay, then x = y.
Similarly, we have the following property for logarithms:
If log x = log y, then x = y.
Example: Solve log3(5x – 6) = log3(x + 2) for x.
Solution:
log3(5x – 6) = log3(x + 2)
5x – 6 = x + 2
5x –x = 2 + 6
4x = 8
x = 2
Example: Solve 2x + 1 = 8 for x.
Solution: Here, the bases are not the same, but we find that we are able to manipulate the right-hand side to make the bases the same.
2x + 1 = 8
2x + 1 = 23
x + 1 = 3
x = 2
Cancellation Laws
Since the exponential and logarithmic functions are inverse functions, cancellation laws apply to give:
loga(ax) = x for all real numbers x
alogax = x for all x > 0
We know that e is the most convenient base to work with for exponential and logarithmic functions. The same cancellation laws apply for the natural exponential and the natural logarithm:
In(ex) = x for all real numbers x
eIn x = x for all x > 0
These last two cancellation laws will be especially useful if you study calculus. To solve a simple exponential equation, you can take the natural logarithm of both sides. (Technically, you can take the logarithm with any base, but the natural log is often the easiest). Similarly, to solve a simple logarithmic equation, you can take the natural exponential of both sides. At this point, the equation can be solved using basic algebra.
Example: Solve e2x = 8 for x.
e2x = 8
In(e2x) = In(8)
2x = In(8)
x = In(8)/2
Example: Solve In(x + 5) = 4 for x.
Solution:
In(x + 5) = 4
eIn(x + 5) = e4
x + 5 = e4
x = e4 - 5
Example 1:
Example 2:
Example 3:
