In this lesson, our instructor Vincent Selhorst-Jones discusses the solving exponential and logarithmic equations. He discusses one to one property, inverse property, solving by inverses and extraneous solutions.
Now that we understand how exponents and logarithms work, how do we solve equations involving them? There are two main ideas that allow us solve such equations: the one-to-one property and the inverse property.
The one-to-one property points out that exponential and logarithmic functions are both one-to-one, that is, different inputs always produce different outputs.
ax = ay ⇔ x = y logb x = logb y ⇔ x=y
If an equation is set up in one of the formats above, we can turn it into something fairly easy to solve.
The inverse property points out that exponentiation and logarithms are inverses of each other: if they have the same base, they "cancel out".
loga (ax) = x blogb x = x
This means we can make both sides of an equation exponents or take the logarithm of both sides to "cancel out" things that are "in the way" of solving an equation.
One particularly useful property of logarithms is
loga xn = n ·loga x.
This allows us to "bring down" exponents with any logarithm. Furthermore, we can use logarithm bases that we have on our calculators to make calculation easy (e⇒ ln, 10 ⇒ log).
Many of the properties we've discussed about exponents and logarithms in previous lessons can be useful in solving exponential or logarithmic equations. If a problem is complicated, try to figure out if you can first simplify it with some of the various
properties we've learned.
While solving these equations, it's important to watch out for extraneous solutions: values that appear over the course of solving, but are not actually solutions. A value might seem like a solution, but actually be outside of the allowed domain.
Make sure to check your answers to be sure they do not cause any issues.
Solving Exponential and Logarithmic Equations
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.