155. Which is bigger, πᵉ or e^π?
Calculators have taken some of the mystery out of this once-challenging question.
(Go ahead and check; you will see that it is a very close call.)
You can answer the question without a calculator, though.
a. Find an equation for the line through the origin tangent to the graph of
y = ln(x).

Verified step by step guidance

1. Identify the function given: \(y = \ln(x)\).
2. Find the derivative of \(y = \ln(x)\) to get the slope of the tangent line at any point \(x\). The derivative is \(y' = \frac{1}{x}\).
3. Since the tangent line passes through the origin (0,0), find the point of tangency \(x = a\) where the tangent line touches the curve \(y = \ln(x)\).
4. The slope of the tangent line at \(x = a\) is \(\frac{1}{a}\), and the tangent line passes through \((a, \ln(a))\) and the origin \((0,0)\).
5. Use the point-slope form of the line through the origin with slope \(\frac{1}{a}\): \(y = \frac{1}{a} x\). This line is tangent to \(y = \ln(x)\) at \(x = a\).

Key Concepts

Comparing Exponential Expressions

To compare expressions like π^e and e^π without a calculator, use logarithms to transform the comparison into a difference of products involving natural logs. This approach leverages properties of exponents and logarithms to analyze which value is larger.

Tangent Line to a Curve

A tangent line to a curve at a point touches the curve without crossing it locally and has the same slope as the curve at that point. For y = ln(x), the slope at any x is given by the derivative 1/x, which helps find the equation of the tangent line.

Derivative of the Natural Logarithm Function

The derivative of y = ln(x) is y' = 1/x, which gives the slope of the tangent line at any point x > 0. This derivative is essential for finding tangent lines and understanding the behavior of the logarithmic function.