Understanding these properties is crucial for manipulating and solving logarithmic equations.
This property relates exponents and logarithms directly:
Simply put: Raising a base ($a$) to the logarithm of a number ($b$) with the same base ($a$) results in the number itself ($b$). The base and the $\log$ effectively “cancel” each other out.
If two logarithms with the same base are equal, their arguments must also be equal:
Simply put: If $\log_a(\text{something}) = \log_a(\text{something else})$, then $\text{something}$ must equal $\text{something else}$. This is a crucial tool for solving logarithmic equations.
The best way to understand any function is to see its graph. Let’s explore the function $y = \log_2 x$ by finding a few key points.
The equation $y = \log_2 x$ is equivalent to $x = 2^y$. This inverse relationship is what we use to find our points.
| x Value | Calculation | y Value | Point (x,y) |
| 1 | $y = \log_2 1 = 0$ | 0 | (1, 0) |
| 2 | $y = \log_2 2 = 1$ | 1 | (2, 1) |
| 4 | $y = \log_2 4 = \log_2 2^2 = 2 \log_2 2 = 2$ | 2 | (4, 2) |
| $\mathbf{\frac{1}{2}}$ | $y = \log_2 \left(\frac{1}{2}\right) = \log_2 2^{-1} = -1$ | -1 | ($\frac{1}{2}$, -1) |
| $\mathbf{\frac{1}{4}}$ | $y = \log_2 \left(\frac{1}{4}\right) = -2$ | -2 | ($\frac{1}{4}$, -2) |
As you can see from the calculated points and the graph provided, the function $y = \log_2 x$ has some distinct characteristics:
The x-intercept is (1, 0). This is a defining characteristic of all basic logarithmic functions, as $\log_b 1 = 0$ for any valid base $b$.
The curve passes through the point (2, 1). Since the base is 2, $\log_2 2 = 1$.
The graph rises slowly as $x$ increases (it’s an increasing function).
It has a vertical asymptote at $x = 0$ (the $y$-axis). The function is undefined for $x \le 0$, meaning the graph approaches the $y$-axis but never touches or crosses it.
The domain (all possible $x$ values) is $x > 0$.
The range (all possible $y$ values) is $(-\infty, \infty)$.
Mastering logarithms starts with understanding these basic properties and their graphical representation. Keep practicing, and you’ll find they are a powerful tool in your mathematical toolkit!