When solving exponential equations, we can sometimes rewrite both sides of the equation with the same base and set the exponents equal to each other. These types of equations are not very challenging to solve, but most of the time we won't be able to rewrite both sides with the same base.
For example, what about 5^x = 26? We can't really rewrite 26 with a base of 5....
The idea is that we need a way to get x out of the exponent so that we can solve for it. In order to do this, we need the inverse operation of exponential functions (Remember the inverse operation "undoes" the operation already there. For example, adding and subtracting are inverse operations because +x and -x undo each other. In the same way, multiplication and division are inverse operations because they undo each other.)
For example, what about 5^x = 26? We can't really rewrite 26 with a base of 5....
The idea is that we need a way to get x out of the exponent so that we can solve for it. In order to do this, we need the inverse operation of exponential functions (Remember the inverse operation "undoes" the operation already there. For example, adding and subtracting are inverse operations because +x and -x undo each other. In the same way, multiplication and division are inverse operations because they undo each other.)
The inverse of the exponential function is the logarithmic function.
Recall that the graphs of two inverse functions are symmetric with respect to y = x.
The graph on the left shows 3 common exponential function and their inverses. Notice their symmetry and common characteristics.
Recall that the graphs of two inverse functions are symmetric with respect to y = x.
The graph on the left shows 3 common exponential function and their inverses. Notice their symmetry and common characteristics.
Before we can take the inverse operation of each side of 5^x=26 in order to solve it, there are a few things we need to know about logarithms first. We'll spend the next few days discussing very important properties of logarithms before returning to solve this problem.
(Note: It is important that you fully understand these properties instead of relying on your calculators for this unit.)
(Note: It is important that you fully understand these properties instead of relying on your calculators for this unit.)
Basics of Logarithms
Part of my passion for logarithms most likely comes from my high school Algebra II teacher. He was most definitely the most demanding and strict teacher I had in high school (Think no free days...ever! Even that one time a tree fell on him...he still came and lectured!). Anyway, he went crazy one day yelling at us the definition of logarithm and, of course, it stuck! I still remember it to this day...
The solution to a logarithm is the exponent to which a base must be raised to produce a given number.
The solution to a logarithm is the exponent to which a base must be raised to produce a given number.
The first thing we need to understand is how to convert between exponential and logarithmic form. Watch the video below to see a few basic examples of how this is done.
Click here to practice evaluating logarithmic expressions on Khan Academy's interactive practice website. Then click here to try a few that are more challenging.
Because exponents and logarithms are related in this manner, there are some very basic properties of logarithms that we like to see happen in our problems!
Because exponents and logarithms are related in this manner, there are some very basic properties of logarithms that we like to see happen in our problems!
It's super easy to prove these properties if you translate each to their exponential form!
Special Logarithms
There are some bases for logarithms that are more common than others. These special logarithms have unique notation.
Common Logarithm - a log with base 10
Whenever you see "log" without a specified base, you can assume that it has a base of 10
Example: log 100 = 2 because 10^2 = 100.
Natural Logarithm - a log with a base e (Euler's number approximately 2.718281828459...)
Whenever you see "ln" this indicates a natural log understood to have a base e. (Why use "ln" instead of "nl?")
Example: ln e^2 = 2 because their common bases cancel.
These are the same "LOG" and "LN" buttons that you see on your calculator. When you use these buttons, your calculator is assuming common log (base 10) and natural log (base e) respectively.
(Side note: This is also what Howard was talking about in his song to Bernadette from earlier posts!)
These special logarithms are the most common to encounter in real life situations. For example,
The assignment for 2/21/14 was Page 178 #1-10, 46, 48. Due Monday!
Common Logarithm - a log with base 10
Whenever you see "log" without a specified base, you can assume that it has a base of 10
Example: log 100 = 2 because 10^2 = 100.
Natural Logarithm - a log with a base e (Euler's number approximately 2.718281828459...)
Whenever you see "ln" this indicates a natural log understood to have a base e. (Why use "ln" instead of "nl?")
Example: ln e^2 = 2 because their common bases cancel.
These are the same "LOG" and "LN" buttons that you see on your calculator. When you use these buttons, your calculator is assuming common log (base 10) and natural log (base e) respectively.
(Side note: This is also what Howard was talking about in his song to Bernadette from earlier posts!)
These special logarithms are the most common to encounter in real life situations. For example,
- Earthquakes use the Richter Scale modeled by M = log (A+B) where A is the amplitude measured by the Seismograph and B is the distance correction factor.
- Sound is measured in decibels (dB for short) by dB = 10 log(p*10^12) where p represents the sound pressure
- Acidity (or Alkalinity) is measured in pH where pH = -log [H+]
The assignment for 2/21/14 was Page 178 #1-10, 46, 48. Due Monday!