Jennifer has an MS in Chemistry and a BS in Biological Sciences.
Let's start with a basic review of logarithms. The logarithm (log) of a number is the exponent a fixed number must be raised to in order to equal a given number. In mathematical terms:
Where b is the base, a is the set number, and c is the given number.
Log functions are important in many areas of science, business, and engineering. For example, the Richter scale that measures the intensity of earthquakes is a logarithmic scale. The growth of bacteria is measured using log functions. And stock brokers can use log functions to predict the growth of a stock portfolio.
The Change of Base Formula
Often, logarithmic equations are written with a base that cannot be easily calculated. Scientific calculators are designed to calculate logs that have a base of ten. Solutions to logs with other bases can be found using charts, or simple calculations. Some are easy, like in the example above. The base is 3, and the problem is asking 'to what exponent must 3 be raised in order to equal 9?' And of course, the answer is 2. Others can be more difficult, like the following:
When converted to exponential forms, this equation becomes 7^x = 13.This problem is impossible to do without looking through a myriad of tables or guessing a thousand (or more) times until you got close.
The easiest way to solve a problem like this is to use the change of base formula. It will allow you to convert the base of any logarithm to something more usable. Most often, you will use it to convert the base to 10, since this is what your calculator uses. The change of base formula is as follows:
To use this to solve the example problem, we can plug in the numbers to get this equation:
Since the logs are now to the base 10, you can use your calculator to solve.
1.1134/0.845 = x
x = 1.318
Which means that 7^1.318 = 13
As with many problems in mathematics, there is a way to check your answer. Just perform the exponent calculation that results from finding the log, and if you get the correct answer, your problem is correct.
For this example, if you plug 7^1.318 into your calculator, you should get 13 - or a number very close, that easily rounds to 13.
Let's try another example: log_4(25) = x
Since it's difficult to solve a log with the base of 4, we can use the change of base formula to convert it. log(25)/log(4). This becomes 1.3979/0.6021
2.3217 (When logarithms with a base of 10 are written out, the 10 is usually understood and not written.)
There is a proof of the change of base formula. The mathematics to show how it works are actually quite simple, although it can look messy at times.
Start by calling the log we're looking for y. This gives us log_a(x) = y. Now we write that in exponential form: x = a^y.
Now we want to solve this equation for y, using only base b logs, not base a logs. To do this, we take the log of each side: log_b(x) = log_b(a^y).
Now we simplify the right side: log_b(x) = y log_b(a).
To get y by itself, we just have to divide both sides by log_b(a), which looks like: log_b(x) / log_b(a) = y.
Substituting log_a(x) back in for y we have: log_a(x) = log_b(x) / log_b(a).
The change of base formula is an easy way to solve logarithms that have a base other than 10. This formula allows you to use your calculator, which is programmed to solve logarithms with base 10, instead of relying on complex mathematical operations or using a table to find the answer.
Once you are finished with this lesson you should be able to:
- State the change of base formula for logarithms
- Use the change of base formula to solve a logarithm with a base other than 10
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