solving logarithmic equations

How to Solve for Logarithms

What are Logarithmic Equations?

Whether you are studying logarithms in your high school math class or preparing for a standardized test such as the ACT or SAT, this particular concept is often one of the most challenging you will encounter in math. A logarithm is an inverse function of an exponentiated number. That confusing definition means that a logarithm is the reverse notation of what you normally see when you calculate an exponent. The combination of the complexity of the concept with the mathematical difficulty of the calculation is what causes students to have such trouble with logarithms.

Since logarithms are just another way of notating exponents, you should begin by reviewing the basics of what exponents are and how they work. An exponent represents how many times a number (or variable like x) is multiplied by itself. At its simplest form this means that, for example, 53 = 5*5*5. While that particular example will give you a real number answer (125), often students encounter exponents in algebra equations with variables, such as y = 2x4 – 6x2 – 56.

Exponents are structured such that you know the number or variable which you are multiplying by itself, also known as the base, and you know the number of times you are multiplying it by itself, the exponent. Your goal is to find the result of that calculation: in other words, the answer. Logarithms, in contrast, work a little differently. In a logarithm, you know the value of the answer and the base, but generally do not know the value of the exponent. That is the point of a logarithm: to find the value of the exponent.


Understanding Logarithmic Notation

The most important aspect to understanding logarithms, even before you start using them to solve advanced math problems, is to recognize how to translate between simple exponential form and the equivalent logarithm. When you have a base of b and you apply an exponent x to is, you find an answer a, like below.

            bx = a

This same equation translates into a logarithm like this:

            logb(a) = x

In the logarithmic notation, the base value is expressed as a subscript next to the notational term “log”. The result or answer to the basic exponent problem appears in parentheses next to the base, while the actual exponent is the solution to the logarithm. Using real numbers, the problem works like this:

            53 = 125 —> log5(125) = 3

How to Solve Logarithmic Equations

To solve basic logarithm problems you can use you ability to translate back and forth between the two formats. Attempt to solve for n in the next example.

            log3(729) = n

Since this translates into 3n = 729, you can use any of a number of strategies or tools to solve this. You can plug the numbers into a calculator or use a “guess and check” strategy of plugging in small numbers, starting with 2, until you achieve a result of 729. Either way, you can figure out that n should equal 6.

In a high school math class such Algebra 2 or pre-Calculus, you would likely encounter such a problem during the first few lessons on logarithms. The questions do get more challenging, though. Math classes include logarithms because of how often the concept is utilized in higher level math, physics, computer science, engineering, and other sciences. There are several concepts that you encounter regularly that are based on logarithms, even if you were not aware of it. Decibels is one example, the unit of measure of the power and amplitude of sound, what we inaccurately call volume. The pH factor of various chemicals, such as water, is also a logarithm.

Once you progress past the basics, logarithms become more complicated as they represent more challenging equations and solutions. Using logarithms, you can accurately calculate negative and factional exponents, which are more difficult to do in the simple exponential form.


Logarithms on the SAT/ACT

On standardized tests such as the ACT and SAT, you will only be expected to remember the basic rules of logarithms. In fact, logarithms are very rare on the SAT, showing up more often on the ACT math section. On the ACT, basic logarithms show up at least once per testing. The only more advanced logarithm concept that shows up every so often is the logarithm of a product.

            logb(a*n) = logb(a) + logb(n)

This reflects that bx1 * bx2 = bx1+x2 in the basic exponent rules. In real numbers, it would look like this:

            log4(16384) = log4(64) + log4(256)

This works because log4(64) = 3 and log4(256) = 4 and 64*256 = 16384.  In exponential form, you can find the same information.

            43 * 44 = 43+4 = 47 = 16384

While this equation is commonly used in a variety of sciences and math applications, it is also a rule of logarithms taught early in the process by math teachers in school. As such, it falls under the heading of the basics of the concept, which is usually from where the ACT test writers draw to build their math questions.

There are numerous other more complicated logarithm problems you will encounter in math class in school, however this is the extent of what you will confront on the ACT. At the very least, this should get you started on the basics in math class and set you up for success on any high level ACT math problems in your future.