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.

April 19, 2025

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

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

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

3. Simplify:

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

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

-3

4. Change of base:

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

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

5. Solve:

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

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

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

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

10. Solve:

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

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

+-sqrt(13)

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Free Math Practice tests on various math topics

Sharpen your math skills with interactive exam‑style questions. Our free math practice tests based on NAPLAN and NSW Curriculum provide instant scoring and detailed explanations so you know exactly where to improve. Whether you’re preparing for school exams or just want to stay ahead, we’ve got you covered.

Attempt the questions on the following math topics. Click on Submit. Get your results. The right answers will be highlighted with explanations in green, whereas wrong answers will be highlighted in red.

Addition

Subtraction

Multiplication

Division

Fractions

BODMAS

Ordering Decimals

Algebra Substitution

Simple Interest

Pythagoras Theorem

Number Plane

Geometry Shapes

Simple Linear Equations

Parabola

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Logarithms

Why You’ll Love These Math Practice Tests

Interactive & Exam‑style

Each test mimics real exam conditions with multiple choice questions, timers and instant feedback. Practise until you’re confident.

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From basic arithmetic to trigonometry and logarithms, our practice tests cover the topics that matter most to students in Years 1–10.

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

What are Logarithms?

Logarithm Formulae and Practice Questions