Question

Match the graph with the function. $$y=\log _{5} x$$

   Match the graph with the function.
$$y=\log _{5} x$$

Precalculus

Cynthia Young 2nd Edition

Chapter 3, Problem 67

Instant Answer

Solved by Expert Linh Vu

06/27/2020

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Step 1: The function given is $$y=\log _{5} x$$ This is a logarithmic function with base 5. Show more…

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Match the graph with the function. $$y=\log _{5} x$$

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Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. They describe the power to which a specific base must be raised to produce a given number. This concept is fundamental in understanding how exponential growth and decay can be 'inverted' to solve for exponents in various mathematical contexts.

Base of a Logarithm

The base of a logarithm, in this case 5, determines the rate at which the function grows or decays. A base greater than 1 results in an increasing function where the output increases as the input grows, which is important for determining the overall shape and behavior of the graph.

Domain and Range

Logarithmic functions have a domain that includes only positive real numbers because you can only take the logarithm of a positive number. Their range, however, is all real numbers, reflecting that the function can output any value from negative to positive infinity, which directly influences the graph's spread.

Graphical Behavior

The graph of a logarithmic function typically passes through the point (1, 0), since any logarithm of 1 is zero. Additionally, the graph increases gradually and shows a vertical asymptote along the y-axis, indicating that the function approaches, but never touches, the line x = 0.

Inverse Relationship with Exponential Functions

Understanding that logarithmic functions are the inverses of exponential functions is crucial. This inverse relationship means that if you have an exponential function with a certain base, the corresponding logarithmic function will effectively 'undo' that operation, which is a key concept in solving equations involving exponentials and logarithms.

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