Question

MAKING AN ARGUMENT Your friend states that every logarithmic function will pass through the point \((1,0)\). Is your friend correct? Explain your reasoning.

Short answer

Yes, your friend is correct. Every basic logarithmic function will pass through the point \((1,0)\).

Step-by-step solution

Understanding Basic Logarithmic Functions

Logarithmic function, is any function of the form \(y = log_b(x)\) where b is a positive real number and not equal to 1, is the base and x is a positive real number. The point (1,0) is a characteristic point of many functions, including logarithmic functions.

Graph of the Logarithmic Function

When graphing the basic logarithmic function \(y = log_b(x)\), regardless of the base b, all graphs will pass through the point (1,0). This is because any number raised to the power of 0 (except 0 itself) is equal to 1, and thus the log of 1 to any base is 0. So, regardless of the value of b (as long as it's a valid base), the graph of \(log_b(x)\) will always pass through (1,0).

Final Verdict

Based on the property and graphical representation of logarithmic functions, every logarithmic function does indeed pass through the point (1,0). Therefore, the friend's statement is correct.

Key concepts

Characteristic Points

Logarithmic functions have some unique features that are helpful to recognize when analyzing their graphs. One such feature is the occurrence of characteristic points. These are specific points which appear as constants across different logarithmic functions. One of the most significant characteristic points in logarithmic functions is the point \(1,0\).
The reason why \(1,0\) is a characteristic point of logarithmic functions is due to the properties of exponents. If you take any positive number \(b\) as a base and raise it to the power of \(0\), the result is always \(1\). This is true for any valid base \(b\) in logarithmic functions like \log_b(x)\. Consequently, the logarithm of \(1\), irrespective of the base, results in \(0\).
This feature simplifies the understanding of how logarithmic functions behave, as it provides a stable point for graphing and analyzing the function across different bases.

Graph of Logarithmic Function

Visualizing logarithmic functions can help tremendously in understanding their behavior. The graph of a basic logarithmic function, expressed as \(y = \log_b(x)\) where \(b\) is any positive number other than \(1\), possesses certain characteristics. One of its defining features is that it consistently passes through the point \(1,0\).
The curve of the logarithmic function approaches the vertical axis asymptotically. The reason behind this is that as \(x\) approaches zero from the positive side, the logarithm becomes negative without bound, depicting the vertical asymptote at \(x = 0\).
Conversely, as \(x\) increases, the logarithmic function increases, albeit at a decreasing rate. This slow increase is why the logarithmic graph maintains its appearance of being more stretched out along the x-axis.
The consistent presence of the point \(1,0\) across all logarithmic graphs, no matter the base, underscores the uniformity in their foundational behavior, which can make function analysis more convenient and predictive for students.

Properties of Logarithms

Understanding logarithms is made easier by familiarizing yourself with a set of properties that logically derive from its definition as an inverse of exponentiation. These properties aid in simplifying complex logarithmic expressions and facilitate solving logarithmic equations.

• Product Rule: \(\log_b(xy) = \log_b(x) + \log_b(y)\). This property indicates that the logarithm of a product is the sum of the logarithms.
• Quotient Rule: \(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\). This expresses that the logarithm of a quotient is the difference of the logarithms.
• Power Rule: \(\log_b(x^n) = n \cdot \log_b(x)\). This property shows that the logarithm of a power is the exponent times the logarithm of the base.
• Change of Base Formula: \(\log_b(x) = \frac{\log_k(x)}{\log_k(b)}\). This useful formula allows finding a logarithm using another base.

These properties are crucial for manipulating and simplifying logarithmic functions in various mathematical problems. They highlight the versatility and usefulness of logarithms in problem-solving and analysis, offering analytical techniques that leverage their defined behaviors.