Question

Graph the given function. $$ f(x)=\log _{5}(x+1) $$

   Graph the given function.
$$
f(x)=\log _{5}(x+1)
$$

Finite Mathematics and Applied Calculus

Finite Mathematics and Applied Calculus

Stefan Waner, Steven… 5th Edition

Chapter 9, Problem 16

Instant Answer

Solved by Expert Gregory Higby

10/02/2021

Step 1

The argument of the logarithm function, $(x+1)$ in this case, must be greater than zero. So, we solve the inequality $x+1 > 0$ to find the domain of the function. This gives us $x > -1$. Show more…

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Graph the given function. $$ f(x)=\log _{5}(x+1) $$

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Key Concepts

Logarithmic Functions

A logarithmic function is the inverse of an exponential function and is defined by the equation f(x) = log_b(x), where b is a positive number not equal to 1. These functions have a domain of positive real numbers (x > 0) for the basic form and exhibit a vertical asymptote at x = 0. Their behavior (increasing or decreasing) is determined by the value of the base; for bases greater than 1, the function increases, while bases between 0 and 1 result in a decreasing function.

Function Transformations

Transformations involve shifting, stretching, or reflecting the graph of a function. In the function f(x) = log_5(x+1), the expression (x+1) indicates a horizontal translation. Specifically, the graph of the base logarithmic function is shifted 1 unit to the left, altering the location of the vertical asymptote from x = 0 to x = -1, while preserving the overall shape of the graph.

Graphical Features of Logarithmic Functions

When graphing logarithmic functions, key features include the vertical asymptote, the x-intercept, and the general increasing or decreasing nature of the curve. The vertical asymptote represents a line that the graph approaches but never touches, and for the given function, it is shifted to x = -1 due to the horizontal translation. Additionally, understanding these features helps in sketching the behavior of the graph accurately over its domain.

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