Math Tips: Exponents and Logs

Sorry it’s been awhile since I’ve posted my math tips – it’s always a little crazy getting through the holidays. I can’t believe in a couple of months we’ll be finished with yet another school year 😉

She Loves Math continues to attract new viewers; I’m getting 1500-2000 hits a day now.  I’ve loved hearing from you and trying to help you solve math problems.

I just completed the Exponential and Logarithmic Functions section; it took me a little longer than expected – sorry about that.  Here is a short article I wrote on exponents and logs for someone; hope it’s useful:

What Is an Exponent?

First, we first need to understand what an exponent is. An exponent just means that a number is multiplied that many times by itself.

Let’s use the following example. If your savings account gives you interest at 5% every year (a rarity, these days!), every year your savings would be multiplied by 1.05. This is because you get to keep what you originally put in your account, your principle (let’s say $100), and add 5% (or .05) times your amount every year. After the first year, your account would have 100(1.05) or $105. After the second year, your account would have 100(1.05)(1.05) or \(100{{\left( {1.05} \right)}^{2}}\), which is $110.25. Do you see how, in the nth year, your account would have \(100{{\left( {1.05} \right)}^{n}}\) dollars?  (“n” is the exponent in this case.)

It’s a confusing concept, but if you do the math year-by-year, it makes sense. By the way, this concept of multiplying your principle by an interest amount every year is called compounding. (Start saving and investing your money early in life; it pays!)


How Do Exponents Relate to Logs?

Now, let’s get back to logs. Logs are related to exponents; in fact, the log of a number is an exponent. Look at the following graphic to see how exponents and logs are related. Remember that every exponent and log has a base, as designated by “b” in this diagram. Our base in the example above is 1.05.

Exponential Function                         Logarithmic Function

\(x={{b}^{y}}\)                               \(\Leftrightarrow \)                              \(y={{\log }_{b}}x\)

\((b>0,\,\,b\ne 1,\,\,x>0)\)

Example:                                                           Example:

   \(\displaystyle 25={{5}^{2}}\)                                                            \(\displaystyle 2={{\log }_{5}}25\)


Note: If there is no “b” next to the log, then the base is assumed to be 10.

Why Do We Need Logs?:  One Example

We are going to show an example of using logs in algebra (the branch of mathematics where letters represent values). But why would we ever want to change an exponential algebraic expression or equation to a logarithmic one? One important purpose of logs is for solving mathematical equations with unknowns or variables in an exponent.


Let’s say we had a beginning amount in our back account ($100) and had an ending account (say, $250), and we wanted to know how many years it took to reach that amount?  We could use the “guess and check” method to try different numbers in the exponent to get the $250 amount.  But the easiest way to get the number of years, or the exponent in the equation \(100{{\left( {1.05} \right)}^{n}}\) is to use logs. So a useful application of using logs is to solve for variables, or unknowns, that are in exponents. 

There are certain properties of logs that very helpful in solving equations.  One of the most useful properties is that the log of a number raised to an exponent is equal to that exponent times the log of the number without the exponent: \({{\log }_{b}}\left( {{{x}^{n}}} \right)=n{{\log }_{b}}x\). This is called the Power Property or Power Rule of logs. This is an important property since it helps “get the exponent down” so we can solve for it using algebra.

Solving a Logarithmic Equation Using Algebra

Let’s go ahead and use logs, including the Power Property, to solve the algebraic equation \(A=100{{\left( {1.05} \right)}^{n}}\) to get the number of years that it would take to go from $100 to $250 with a yearly interest rate of 5%. We want A (the ending amount) equal to $250:

So the number of years it would take to have $250 in your account is 18.78 years. This method is much easier than the “guess and check” method that involves just trying different numbers in the exponents and seeing if you get the $250.  And we can check our answer by using our calculator: \(A=100{{\left( {1.05} \right)}^{{18.78}}}\approx 250\).

And this is just one of many applications of using logs; they are also used in biology (population growth), anthropology (carbon dating), geology (the Richter scale for earthquakes), chemistry (pH scales) and many other fields.

So That’s Why We Need Logs!

Logs are here to stay and are extremely powerful in mathematics. They are used in everyday applications of science and finance; in fact, logs are used at your local bank to help with the interest in your bank. And again you can refer to the Exponential and Logarithmic Functions section of She Loves Math to learn more!

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