Logarithms - Decimals, Fractions & Tricky Cases

(calculating logarithms by hand - Part 2)


Now that we know how to calculate "basic" logarithms, we learn how to calculate more tricky logarithms, involving decimals, fractions and improper fractions.

For instance, we need to know how to calculate \(log_4(0.25)\), \(log_5\begin{pmatrix}\frac{1}{25}\end{pmatrix}\), \(log_{25}(75)\) or even \(log_{81}\begin{pmatrix}27\end{pmatrix}\).

We learn about three distinct scenarios that will teach us how to evaluate each of these types of logarithms.


Scenario 1

At times we'll have to evaluate logarithms \(log_b\begin{pmatrix}a\end{pmatrix}\) for which the input \(a\) is a decimal or a fraction. An example could be: \[log_{4}(0.25)\]

Tutorial 1

Exercise 1

Calculate each of the following without a calculator:

  1. \(log_{10}(0.1)\)
  2. \(log_3\begin{pmatrix}\frac{1}{27}\end{pmatrix}\)
  3. \(log_2(0.5)\)
  4. \(log_4\begin{pmatrix}\frac{1}{64}\end{pmatrix}\)
  5. \(log_2(0.25)\)
  6. \(log_{10}\begin{pmatrix}\frac{1}{10000}\end{pmatrix}\)
  7. \(log_4(0.25)\)
  8. \(log_9\begin{pmatrix}\frac{1}{81}\end{pmatrix}\)
  9. \(log_5(0.2)\)
  10. \(log_{10}(0.001)\)

Note: this exercise can be downloaded as a worksheet to practice with: worksheet

Answers Without Working

  1. \(log_{10}(0.1) = -1\)
  2. \(log_3\begin{pmatrix}\frac{1}{27}\end{pmatrix} = -3\)
  3. \(log_2(0.5)= -1\)
  4. \(log_4\begin{pmatrix}\frac{1}{64}\end{pmatrix} = -3\)
  5. \(log_2(0.25)= - 2\)
  6. \(log_{10}\begin{pmatrix}\frac{1}{10000}\end{pmatrix} = -4\)
  7. \(log_4(0.25) = -1\)
  8. \(log_9\begin{pmatrix}\frac{1}{81}\end{pmatrix} = -2\)
  9. \(log_5(0.2) = -1\)
  10. \(log_{10}(0.001)=-3\)


Scenario 2

At times we'll have to evaluate logarithms \(log_b\begin{pmatrix}a\end{pmatrix}\) for which the base \(b\) is greater than the input \(a\). An example could be: \[log_{32}(4)\]

Tutorial 2: when the base is greater than the input

Exercise 2

Calculate each of the following without a calculator:

  1. \(log_4(2)\)
  2. \(log_8(2)\)
  3. \(log_{81}(3)\)
  4. \(log_{81}(9)\)
  5. \(log_{64}(4)\)
  6. \(log_{49}(7)\)
  7. \(log_{36}(6)\)
  8. \(log_{125}(5)\)
  9. \(log_{10000}(10)\)
  10. \(log_{625}(25)\)

Note: this exercise can be downloaded as a worksheet to practice with: worksheet

Answers Without Working

We find the following:

  1. \(log_4(2) = \frac{1}{2}\)
  2. \(log_8(2) = \frac{1}{3}\)
  3. \(log_{81}(3) = \frac{1}{4}\)
  4. \(log_{81}(9) = \frac{1}{2}\)
  5. \(log_{64}(4) = \frac{1}{3}\)
  6. \(log_{49}(7) = \frac{1}{2}\)
  7. \(log_{36}(6) = \frac{1}{2}\)
  8. \(log_{125}(5) = \frac{1}{3}\)
  9. \(log_{10000}(10) = \frac{1}{4}\)
  10. \(log_{625}(25) = \frac{1}{2}\)


Scenario 3

At times we'll have to evaluate logarithms \(log_b\begin{pmatrix}a\end{pmatrix}\) for which the base \(b\) and the input \(a\) aren't "direct" powers of each other. An example could be: \[log_9(27)\] we can see that \(27\) isn't a "simple" power of \(9\), nor is \(9\) a simple power of \(27\).

Tutorial 3


Exercise 3

Calculate each of the following, without using a calculator:

  1. \(log_{125}(625)\)
  2. \(log_{81}(27)\)
  3. \(log_{9}(27)\)
  4. \(log_{64}(128)\)
  5. \(log_{216}(36)\)
  6. \(log_{128}(16)\)
  7. \(log_{8}(4)\)
  8. \(log_{36}(216)\)
  9. \(log_{100000}(100)\)
  10. \(log_{8}(128)\)

Note: this exercise can be downloaded as a worksheet to practice with: worksheet

Answers Without Working

We find:

  1. \(log_{125}(625) = \frac{4}{3}\)
  2. \(log_{81}(27) = \frac{3}{4}\)
  3. \(log_{9}(27) = \frac{3}{2}\)
  4. \(log_{64}(128) = \frac{7}{6}\)
  5. \(log_{216}(36) = \frac{2}{3}\)
  6. \(log_{128}(16) = \frac{4}{7}\)
  7. \(log_{8}(4) = \frac{2}{3}\)
  8. \(log_{36}(216) = \frac{3}{2}\)
  9. \(log_{100000}(100) = \frac{2}{5}\)
  10. \(log_{8}(128 =\frac{7}{2}\)


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