Question

Graph the function. $$ f(x)=\log _{5}(x+4) $$

   Graph the function.
$$
f(x)=\log _{5}(x+4)
$$

College Algebra Essentials

College Algebra Essentials

Julie Miller 1st Edition

Chapter 4, Problem 109

Instant Answer

Solved by Expert Subhadeepta Sahoo

08/12/2021

Step 1

The function $f(x)=\log _{5}(x+4)$ is a logarithmic function with base 5 and the argument is $x+4$. Show more…

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Graph the function. $$ f(x)=\log _{5}(x+4) $$

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Key Concepts

Logarithmic Functions

A logarithmic function is the inverse of an exponential function and is defined for positive inputs when written in the form f(x) = log_b(x). Such functions are characterized by their slowly increasing or decreasing behavior, depending on the base. With bases greater than one, these functions are strictly increasing and are used to model scenarios where growth rates decelerate over time.

Domain and Vertical Asymptote

The domain of a logarithmic function is determined by the condition that the argument of the logarithm must be positive. For the function f(x) = log_5(x+4), the input to the log function, x+4, must be greater than 0, which establishes the vertical asymptote at x = -4, as the function approaches negative infinity when x approaches this value from the right.

Horizontal Shifts

Horizontal shifts in functions occur when a constant is added to or subtracted from the variable within the function. In this case, the transformation from log_5(x) to log_5(x+4) results in a leftward shift of the graph by 4 units. This shift moves all key features of the standard logarithmic graph, including the asymptote, without altering the shape or the rate of change of the graph.

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