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Logarithms Made Super Easy (With Formulae & a Confidence-Boosting Test!)

Logarithms are essential for students due to their foundational role in higher mathematics, their ability to simplify complex calculations, and their widespread applications in science, technology, and problem-solving. Get easy-to-understand theory, Formulae, and Interactive Exam-Style Questions with Instant Scoring and Detailed Explanations.
Logarithms Made Super Easy (With Formulae & a Confidence-Boosting Test!)

Table of Contents

What are Logarithms?

Logarithms trying to find out the power you have to raise to get figure out another number.

Logarithm Formulae and Practice Questions

Logarithm Formulae and Practice Questions

The main Logarithm Formula is:

If \(b^x = n\), then \(\log_{b}(n) = x\)


Common Logarithm Formulas:

  • \(\log_{b}(1) = 0\)
  • \(\log_{b}(b) = 1\)
  • \(\log_{b}(xy) = \log_{b}(x) + \log_{b}(y)\)
  • \(\log_{b}(x/y) = \log_{b}(x) - \log_{b}(y)\)
  • \(\log_{b}(x^n) = n \log_{b}(x)\)
  • \(\log_{b}(a) = \frac{\log_{c}(a)}{\log_{c}(b)}\)


Logarithm Quiz

To write the answers, Use logb(x^n), sqrt for √, +- for ±.

1. Solve for \(x\) :

a) \(\log_{4}(x) = 2\)

\(\log_{4}(x) = 2 \implies x = 16\)

b) \(\log_{5}(x) = 3\)

\(\log_{5}(x) = 3 \implies x = 125\)

2. Evaluate:

a) \(\log_{2}(16)\)

4

b) \(\log_{3}(81)\)

4

3. Simplify:

a) \(\log_{5}(25)\)

2

b) \(\log_{2}\bigl(\tfrac{1}{8}\bigr)\)

-3

4. Change of base:

a) \(\log_{4}(16)\)

2

b) \(\log_{6}(36)\)

2

5. Solve:

a) \(\log_{3}(x) = 1 + \log_{3}(27)\)

81

b) \(\log_{2}(2x) = 3\)

4

6. Expand:

a) \(\log_{7}(x^2 y)\)

2*log7(x) + log7(y)

b) \(\log_{2}\bigl(\tfrac{x}{y^3}\bigr)\)

log2(x) - 3*log2(y)

7. Condense:

a) \(3\log_{2}(x) - \log_{2}(y)\)

log2(x^3/y)

b) \(\tfrac12\log_{3}(x) + \log_{3}(y)\)

log3(sqrt(x)) + log3(y)

8. Solve:

a) \(\log_{4}(x-1)+\log_{4}(x+1)=2\)

+-sqrt(17)

b) \(\log_{3}(x+2)-\log_{3}(x-2)=2\)

2.5

9. Evaluate:

a) \(\log_{5}(125)\)

3

b) \(\log_{3}(27)\)

3

10. Solve:

a) \(\log_{2}(x)+\log_{2}(x-2)=3\)

4

b) \(\log_{3}(x^2-4)=2\)

+-sqrt(13)

Aditya Jadhav

Aditya Prakash Jadhav, commonly known as Adi, is the founder of Aussie Math Tutor NSW, a private maths tutoring service based in Telopea, NSW. With over 5 years of tutoring experience, he helps students build confidence in maths through patient, step-by-step support aligned with the NSW NESA Mathematics syllabus. He creates student-friendly maths content, practice questions, and learning resources for Years 1–10, including NAPLAN, Selective School and OC preparation. Adi holds postgraduate qualifications in management and marketing, has an IT engineering background, and is also the author of “Study Notes for Management 101,” published by Wiley Publications.