Introduction to Logarithms
📘 What is a Logarithm?
A logarithm answers a very specific question:
To what power must we raise a given base to obtain a certain number?
This relationship is written:
\[\log_a b = x \quad \Longleftrightarrow \quad a^x = b\]We are used to reading powers from left to right:
- For example:
\(2^3 = 8\)
But a logarithm goes in the opposite direction:
- It asks:
\(\log_2 8 = ?\)
and answers:
\(\log_2 8 = 3\)
because \(2^3 = 8\)
⚠️ When is a Logarithm Defined?
A logarithm like $\log_a b$ is only defined under two conditions:
- The base $a$ must be positive and different from 1
- The argument $b$ must be positive
In symbols:
\[a > 0,\quad a \neq 1,\quad b > 0\]Why? Because:
- We can’t raise a negative base to arbitrary powers (e.g., roots)
- $a = 1$ would always give the same result: $1^x = 1$
- The result of an exponential function $a^x$ is always positive, so the inverse (logarithm) is only defined for positive inputs
🧠 A Step-by-Step Example
Let’s say we want to solve this exponential equation:
\[2^x = 5\]We ask: “What power of 2 gives 5?”
There is no integer that works exactly, so we use logarithms:
\[x = \log_2 5\]This is a precise expression, just like $\sqrt{2}$.
Its decimal approximation is:
This means:
“2 raised to the power 2.3219 is approximately 5.”
🔁 Logarithms and Exponentials: Inverse Functions
The logarithm base $a$ is the inverse of the exponential function base $a$:
- Exponential:
\(f(x) = a^x\) - Logarithm:
\(g(x) = \log_a x\)
These functions “undo” each other:
\[\log_a (a^x) = x \qquad \text{and} \qquad a^{\log_a x} = x\]This is similar to how square roots and squaring are inverses:
\[\sqrt{x^2} = x \qquad \text{and} \qquad (\sqrt{x})^2 = x\]📈 Graphical Features of the Logarithmic Function
For the function:
\[f(x) = \log_a x\]we know:
- It is defined only for $x > 0$
- It passes through the point $(1, 0)$, since
\(\log_a 1 = 0\) - It increases if $a > 1$, and decreases if $0 < a < 1$
- It grows slowly: logarithms increase very slowly for large values
A classic example:
\[\log_{10} 1000 = 3 \qquad \text{because} \qquad 10^3 = 1000\]But:
\[\log_{10} 10000 = 4 \quad \Rightarrow \quad \text{just one unit more}\]So even multiplying by 10 gives only a small change in the logarithm.
🔧 Core Logarithmic Rules
These rules are essential for simplifying logarithmic expressions:
Product Rule
\[\log_a (bc) = \log_a b + \log_a c\]Quotient Rule
\[\log_a \left( \frac{b}{c} \right) = \log_a b - \log_a c\]Power Rule
\[\log_a (b^n) = n \cdot \log_a b\]Change of Base Formula
To compute logarithms with a base you don’t have on your calculator:
\[\log_a b = \frac{\log_c b}{\log_c a}\]Most often, we use $\log_{10}$ or $\ln$ (log base $e$).
📝 Practice: A Detailed Example
Let’s simplify this expression:
\[2 \log x + 3 \log y\]We apply the power rule first:
\[= \log(x^2) + \log(y^3)\]Now the product rule:
\[= \log(x^2 y^3)\]This shows how multiple terms can be condensed into a single logarithm.
Another common question:
What is $\log_2 \sqrt[3]{16}$?
We note:
\[\sqrt[3]{16} = 2^{4/3}\]So:
\[\log_2(2^{4/3}) = \frac{4}{3}\]This uses the rule:
\[\log_a (a^x) = x\]🔎 Want to Go Further?
Try proving these properties from the definition:
- Why does the product rule work?
- Can you explain why logarithms grow slowly?
- Explore the graph of $y = \log_a x$ for different values of $a$
And when you’re ready…