Question

In Exercises, determine an equation of the tangent line to the function at the given point. $$ y=\log _{3} x $$ $$ (27,3) $$

Short answer

The equation of the tangent line to the function \(y = \log_3 x\) at the point (27,3) is \(27 \ln 3 (y - 3) = x - 27\)

Step-by-step solution

Determine the derivative of the function

The derivative of the function \(y = \log_3 x\) can be found by using the formula for the derivative of the logarithm base \(a\) of \(x\), which is \(\frac{1}{x \ln a}\). This gives: \(y' = \frac{1}{x \ln 3}\)

Calculate the slope of the tangent at the given point

Substitute the x-coordinate of the given point into the derivative to find the slope of the tangent at that point. The x-coordinate of the given point is 27. So, substitute \(x=27\) into \(y' = \frac{1}{x \ln 3}\) to get \(m= \frac{1}{27 \ln 3}\)

Write the equation of the tangent line

Use the point-slope form of the line equation, which is \(y - y1 = m(x - x1)\), where \((x1, y1)\) are the coordinates of the given point and \(m\) is the slope of the line. Substituting \(m = \frac{1}{27 \ln 3}\), \(x1 = 27\), and \(y1 = 3\) into the equation gives: \(y - 3 = \frac{1}{27 \ln 3}(x - 27)\)

Simplify the equation

If you simplify the equation obtained in the previous step by multiplying all terms by \(27 \ln 3\) to clear the fraction, you get: \(27 \ln 3 (y - 3) = x - 27\)

Key concepts

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions that are composed of logarithms. In our exercise, the function in question is \(y = \log_3{x}\), which is a logarithm with base 3. The derivative of a logarithmic function of the form \(\log_a{x}\) is derived using a specific formula. This formula is \(\frac{1}{x \ln a}\), where \(a\) is the base of the logarithm.

Applying this formula allows us to find the derivative of \(y = \log_3{x}\). By substituting \(3\) for \(a\), we gain the expression \(y' = \frac{1}{x \ln 3}\).

This derivative is particularly useful in calculus when determining the slope of a tangent line at any point on the curve of the logarithmic function. It gives us a tool to measure how steep the curve is at any given \(x\) value, efficiently connecting logarithmic functions to their geometrical properties.

Point-Slope Form

The point-slope form is a method of expressing a linear equation. It's particularly handy when you need to find an equation of a line that passes through a given point with a known slope. The general formula is \(y - y_1 = m(x - x_1)\).

In this formula, \((x_1, y_1)\) are the coordinates of the known point, and \(m\) represents the slope of the line. In the context of the problem given, the point is \((27, 3)\), and the slope \(m\) is calculated from the derivative.

This form is intuitive as it builds directly from a certain point on the line and specifies how far the line moves upward or downward (its slope) as it moves horizontally. After substituting the point and slope into the equation, you get a simplified form, making it easy to see the relation between a point on the line and the slope.

Derivative Formula

The derivative formula is a fundamental concept in calculus that helps us find the rate at which one quantity changes relative to another. In simpler terms, the derivative allows us to understand how a function behaves as its input changes.

The derivative of a function like \(y = \log_3{x}\) is found using the rule that pertains to logarithmic functions. Specifically, the derivative for \(\log_a{x}\) is \(\frac{1}{x \ln a}\). For our example, substituting in the base \(3\), gives \(y' = \frac{1}{x \ln 3}\).

Derivatives are used beyond just calculating the slope of a tangent line—they're essential in many real-world applications, from physics to economics. In this exercise, we use the derivative formula to derive the slope of the tangent line at a specified point \((27,3)\). By inserting the \(x\)-value (27) into the derivative \(y'\), we determine \(m= \frac{1}{27 \ln 3}\). This slope is then used in the point-slope form to find the full equation of the tangent line.