Graphing Logarithmic Functions With Transformations, Asymptotes, and Domain & Range  Summary and Q&A
TL;DR
Learn how to graph and transform logarithmic functions easily using key steps and points.
Questions & Answers
Q: How can we graph logarithmic functions easily?
To graph logarithmic functions, start by setting the inside of the function equal to zero, one, and the base value. Use the solutions to find the vertical asymptote and key points for plotting.
Q: What is the range of logarithmic functions?
The range of logarithmic functions is always all real numbers, from negative infinity to positive infinity. This is because the graph extends indefinitely in the ydirection.
Q: How do transformations affect logarithmic functions?
Applying a negative sign in front of the logarithm reflects the graph over the xaxis. Similarly, applying a negative sign in front of the x reflects the graph over the yaxis. The combination of both reflections results in a reflection over the origin.
Q: How do you determine the domain and range of a logarithmic function?
The domain represents all allowable xvalues for the function and is determined by analyzing the graph from left to right. The lowest xvalue is the vertical asymptote, and the highest extends to positive infinity. The range is always all real numbers, as the graph extends indefinitely upwards.
Summary & Key Takeaways

Start by setting the inside of the logarithmic function equal to zero, one, and the base value.

Use the solved values to determine the vertical asymptote and key points to plot on the graph.

Understand the effects of applying transformations, such as reflecting the graph over the x or y axis or over the origin.

Write the domain and range based on the graph and equation properties.