🔍 Why Do We Need Logarithms?
✅ Equations We CAN Solve Already
Consider the equation:
$2^x = 8$
We recognize that $8 = 2^3$, so we rewrite:
$2^x = 2^3 \implies x = 3$
Easy! The exponent is a nice integer.
❌ Equations We CANNOT Solve (Yet)
Now consider the equation:
$2^x = 5$
We cannot write 5 as $2^{\text{nice number}}$.
$2^2 = 4$ is too small. $2^3 = 8$ is too big.
So $x$ is somewhere between 2 and 3, but what exactly is it?
We need a new mathematical concept!
"To what exponent must we raise a given base to obtain a given number?"
📐 Definition of Logarithm
DEFINITION
The logarithm base $a$ of a positive number $b$ is the exponent to which $a$ must be raised to get $b$.
$$\log_a b = x \quad \Longleftrightarrow \quad a^x = b$$
Conditions: $a > 0$, $a \neq 1$, $b > 0$
📖 Vocabulary
- In $\log_a b = x$: the number $a$ is called the base
- The number $b$ is called the argument
- The result $x$ is the logarithm (i.e., the exponent)
🔢 Examples:
$\log_2 8 = 3$
because $2^3 = 8$
$\log_3 81 = 4$
because $3^4 = 81$
$\log_5 \tfrac{1}{25} = -2$
because $5^{-2} = \tfrac{1}{25}$
🔄 Logarithm and Exponential Are Inverse Operations
The logarithmic form and the exponential form are two ways of writing the same relationship. They are interchangeable:
$$\log_a b = x \quad \Longleftrightarrow \quad a^x = b$$
Logarithmic Form $\longleftrightarrow$ Exponential Form
Converting between these two forms is the most fundamental skill in working with logarithms.
| Logarithmic Form | ↔ | Exponential Form |
|---|---|---|
| $\log_7 49 = 2$ | ↔ | $7^2 = 49$ |
| $\log_{10} 0.001 = -3$ | ↔ | $10^{-3} = 0.001$ |
| $\log_2 32 = 5$ | ↔ | $2^5 = 32$ |
| $\log_4 2 = \tfrac{1}{2}$ | ↔ | $4^{1/2} = 2$ |