3.4. Exponential and Logarithmic Equations
We are often presented with equations involving exponential or logarithmic functions. In this section we show how to solve the simplest of these equations, i.e. equations of one of the two following forms:
- (3.4.1) bx = c
- (3.4.2) logbx = c
Both of these equations are easy to solve using Equations (3.3.6) and (3.3.7) respectively. For instance, taking the base-b logarithm of both sides of (3.4.1) and using (3.3.5) with b=10, we obtain
logb(bx) = logbc = log c / log b.
But by (3.3.6), we see that the left side is just x. Thus we obtain the following solution to (3.4.1):
- (3.4.3) x = logbc.
Exponentiating both sides of (3.4.2) with base b, we obtain
Using (3.3.7), we see once again that the left side is x, whence we have the solution
- (3.4.4) x = bc.
Example 1: Solve the exponential equation 2x = 10.
Solution: The solution is given by x = log210 = log 10 / log 2 = 1 / log 2, which one easily calculates to be approximately 3.32 using a scientific calculator.
Example 2: Solve the exponential equation 0.5x = 0.04.
Solution: The solution is given by x = log0.50.04 = log 0.04 / log 0.5 = 4.64.
Example 3: Solve the logarithmic equation log x = 4.26.
Solution: The solution is given by x = 104.26 ≈ 18,197.
Example 4: Solve the logarithmic equation logπx = 1.76.
Solution: The solution is given by x = π1.76 ≈ 7.50.