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

What are Logarithms?

Logarithm Formulae and Practice Questions