🗺️ Chapter Roadmap
1 What is a Logarithm?
A Logarithm is simply the opposite of an Exponent (Power). While an exponent tells you what a number becomes when raised to a power, a logarithm tells you what power was used to reach a certain number.
The Basic Question:
If someone asks: "How many times should I multiply 2 by itself to get 8?" The answer is 3. In math language, the logarithm of 8 with base 2 is 3.
2 Converting Forms
In exams, you are often asked to switch between these two ways of writing the same thing:
Exponential Form
"Two to the power of three is eight"
Logarithmic Form
"Log of eight to the base two is three"
3 The 3 Laws of Logarithms
These laws allow us to simplify very large numbers into simple additions and subtractions. Memorize these for your test:
1. Product Rule
log A + log B
"Multiplication turns into Addition"
2. Quotient Rule
log A − log B
"Division turns into Subtraction"
3. Power Rule
n × log A
"The power drops to the front"
4 Important Base Rules
In math exams, if the base is not written, it is assumed to be one of these two:
log(100) is same as log10(100) = 2
Written as "ln x"
5 The Change of Base Rule
Calculators usually only have log (base 10) and ln (base e). If you need to find log2(5), you must change the base using this formula:
Example: log2(5) = log(5) / log(2)
💡 Quick Identity Tricks:
- • loga(1) = 0 (Anything to power 0 is 1)
- • loga(a) = 1 (Anything to power 1 is itself)
- • loga(ak) = k