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:
But a logarithm goes in the opposite direction:
- It asks:
and answers:
\[\log_2 8 = 3\]because
\[2^3 = 8\]When Is a Logarithm Defined?
A logarithm like logₐ 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 real powers in general
- a = 1 would always give the same result: 1ˣ = 1
- The result of an exponential function aˣ 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 √2.
Its decimal approximation is:
\[\log_2 5 \approx 2.3219...\]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:
- Logarithm:
These functions “undo” each other:
\[\log_a (a^x) = x\]and
\[a^{\log_a x} = x\]A similar relationship exists between squaring and square roots, but with an important restriction.
For every real number x:
\[\sqrt{x^2} = |x|\]If we restrict the squaring function to x ≥ 0, then the square root is its inverse, and:
\[\sqrt{x^2} = x\]for x ≥ 0.
Likewise:
\[(\sqrt{x})^2 = x\]for x ≥ 0.
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
- 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 log₁₀ 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 log₂ ∛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 = logₐ x for different values of a