Graphs of Logarithmic Functions
Graphs of logarithmic functions is a Grade 7 math skill from Yoshiwara Intermediate Algebra where students analyze the shape, domain, range, and key features of y = log_b(x). The graph passes through (1, 0), has a vertical asymptote at x = 0, and increases slowly for b > 1.
Key Concepts
Property The graph of a logarithmic function $g(x) = \log b x$ can be found by reflecting the graph of its inverse exponential function, $f(x) = b^x$, across the line $y=x$. To create a table of values for $g(x) = \log b x$, you can interchange the columns in a table for $f(x) = b^x$.
Examples Since the point $(3, 27)$ is on the graph of the exponential function $y=3^x$, the point $(27, 3)$ must be on the graph of the logarithmic function $y=\log 3 x$.
The graph of $y=10^x$ passes through $(0, 1)$ and $(1, 10)$. Therefore, the graph of its inverse, $y=\log {10} x$, must pass through $(1, 0)$ and $(10, 1)$.
Common Questions
What does the graph of a logarithmic function look like?
The graph of y = log_b(x) passes through the point (1, 0) and (b, 1), has a vertical asymptote at x = 0, and increases slowly to the right for b > 1.
What is the domain and range of y = log_b(x)?
The domain is x > 0 (all positive real numbers) and the range is all real numbers.
How is the graph of ln(x) different from log(x)?
Both have the same shape. ln(x) uses base e ≈ 2.718 so it rises more steeply, while log(x) uses base 10.
How do transformations affect the graph of a logarithmic function?
Adding/subtracting inside shifts left/right; adding outside shifts up/down; multiplying by a constant stretches or reflects the graph.