Graphs of Logarithmic Functions

Domain and Intercepts

When sketching or identifying the graph of a logarithmic function, there are two critical points to remember regarding its boundaries and intercepts:

1. The Domain (x-values): Logarithmic functions of the form $y = \log_a x$ are only defined for $x > 0$. This means the graph will never exist in the region where $x$ is zero or negative; it will never cross the $y$-axis.
2. The x-intercept: All basic logarithmic functions of the form $y = \log_a x$ pass through the point $(1, 0)$. This is because any base raised to the power of zero equals one ($a^0 = 1$), which in logarithmic form translates to $\log_a 1 = 0$.

Exam Tip: If you are asked to find the equation of a log function and you are given a point on the graph, remember that the $x$-value must be positive. If a graph passes through $(0.1, 10)$, you can use this to solve for the coefficient $a$ in $y = a \log_{10} x$.

Asymptotic Behavior

As the value of $x$ approaches zero from the positive side ($x \to 0^+$), the logarithmic function exhibits a specific type of behavior known as a vertical asymptote.

As $x$ gets closer and closer to zero, the value of $y$ tends towards negative infinity ($-\infty$). Visually, this means the curve will drop sharply, hugging the $y$-axis without ever actually touching or crossing it.

Worked Example: Finding a Function from a Point

In engineering applications, you may need to determine the specific equation of a logarithmic curve based on observed data.

Example: Find the equation of the logarithmic function $y = a \log_{10} x$ given that the graph passes through the point $(0.1, 10)$.

Solution:

1. Substitute the known coordinates $x = 0.1$ and $y = 10$ into the general equation: $10 = a \log_{10}(0.1)$
2. We know that $0.1 = 10^{-1}$, so $\log_{10}(0.1) = -1$: $10 = a \cdot (-1)$
3. Solve for $a$: $a = -10$
4. Therefore, the equation is: $y = -10 \log_{10} x$

Summary of Key Properties

To quickly identify a logarithmic graph, use this checklist:

By understanding these fundamental graphical traits, you can effectively transpose between exponential and logarithmic forms, which is a vital skill for solving maritime and engineering equations.