How to Evaluate Basic Logarithmic Equations?

By Robert O

Logarithms borrow more from exponents. If you have the knowledge of exponents, including laws of exponents, then evaluating any logarithmic equations will not be a problem. In this tutorial, we will discuss how to evaluate basic logarithms without a calculator.

Most scientific calculators will only give you logarithms to base 10 or base e. If you don’t want to use a calculator, then recall your knowledge of squares, cubes, square roots, and exponents. We will apply them here to make it possible to evaluate any basic logarithms.

Logarithm notation

If you have the equation


then the logarithm form is

    \[log _{b} y=x,\]

where b is the base of the logarithm and x is the exponent. In other words, logarithms ask us the number times a given number is multiplied by itself to find another number.

Let’s evaluate a few logarithms to see it in action.

Example 1

Evaluate each of the following logarithms:

    \[\text { a) } log _{2} 4\]

    \[\text { b) } log _{3} 27\]

    \[\text { c) } log _{7} \frac{1}{49}\]

    \[\text { d) } log _{\frac{1}{3}} 81\]

    \[\text { e) } log _{\frac{3}{2}} \frac{27}{8}\]


    \[\text { a) } log _{2} 4=?\]

Using the notation that we already know, we can rewrite the logarithmic form to exponential form.


The problem is now simple to solve without a calculator.



    \[log _{2} 4=2\]

We will use the same concept to evaluate the remaining logarithms.

    \[\text { b) } log _{3} 27=?\]



    \[\log _{3} 27=3\]

    \[\text { c) } log _{7} \frac{1}{49}=?\]


    \[\frac{1}{49}=7^{-2} \text {, applying the explonential law } a^{-x}=\frac{1}{a^{x}}\]

    \[log _{7} \frac{1}{49}=-2\]

    \[\text { d) } log _{\frac{1}{3}} 81=?\]



    \[log _{\frac{1}{3}} 81=-4\]

    \[\text { e) } log _{\frac{3}{2}} \frac{27}{8}=?\]



    \[log _{\frac{3}{2}} \frac{27}{8}=3\]


It is easy to evaluate basic logarithms if you already know the laws of exponents. From the tutorial, logarithms and exponents are two very close concepts that share many things in common. However, it may be difficult to learn logarithms before being well-conversant with exponents. We have another tutorial on exponents that you can look at if you find it difficult to evaluate basic algorithms.

About the Author

This lesson was prepared by Robert O. He holds a Bachelor of Engineering (B.Eng.) degree in Electrical and electronics engineering. He is a career teacher and headed the department of languages and assumed various leadership roles. He writes for Full Potential Learning Academy.