Question

Convert the following expressions to the indicated base. $$\log _{2}\left(x^{2}+1\right) \text { using base } e$$

Short answer

Question: Rewrite the logarithmic expression $\log _{2}\left(x^{2}+1\right)$ using base $e$. Answer: $\frac{\ln(x^2+1)}{\ln(2)}$

Step-by-step solution

Identify the change of base formula

The change of base formula is: $$\log _{b}(a) = \frac{\log K(a)}{\log K(b)}$$

Apply the change of base formula

Using the formula with \(a = x^2 + 1\), \(b = 2\), and \(K = e\), we have: $$\log _{2}\left(x^{2}+1\right) = \frac{\log_e(x^2+1)}{\log_e(2)}$$

Simplify the expression

When using base \(e\), we usually denote it with the natural logarithm function, "ln": $$\frac{\log_e(x^2+1)}{\log_e(2)} = \frac{\ln(x^2+1)}{\ln(2)}$$

Final answer

The expression in base \(e\) is: $$\log _{2}\left(x^{2}+1\right) = \frac{\ln(x^2+1)}{\ln(2)}$$

Key concepts

The Essence of Logarithmic Expressions

Logarithmic expressions are a fundamental tool in understanding mathematical relationships involving exponential growth or decay. A logarithm answers the question: to what exponent must we raise a specific base to obtain a certain number? This concept is written as log_b(a), where b is the base, and a is the number we're interested in. Think of it as the inverse action of exponentiation; if b raised to the y gives us a, then the logarithm of a with base b is y, expressed as log_b(a) = y.

For instance, since 2 raised to the power of 3 equals 8, the logarithm of 8 in base 2 is 3, or log_2(8) = 3. It's important to note that the base of a logarithm can be any positive number other than 1, but two bases are particularly common: 10, used in common logarithms, and e, the irrational number approximately equal to 2.71828, used in natural logarithms.

Understanding the Natural Logarithm (ln)

The natural logarithm is a special logarithmic function where the base is the irrational number e. Denoted as ln(x), it's widely used in various fields such as science, engineering, and specifically in calculus, due to its unique properties. One of the key characteristics that make e and consequently the natural logarithm so special is related to the concept of natural growth and the rate of change.

The formula e^x represents continuous growth, and its derivative, a concept from calculus describing how a function changes at any given point, is itself. Similarly, the derivative of ln(x) is 1/x, making computation in calculus much more manageable. Whether you're calculating compound interest or describing natural phenomena like radioactivity or population growth, the natural logarithm is a crucial tool for modeling these processes mathematically.

Calculus: Connecting Logarithms and Change

Calculus is the mathematical study of continuous change, and it's in this field that logarithms, especially the natural logarithm, truly shine. There are two main branches of calculus: differential calculus, which deals with the rate of change and slopes of curves, and integral calculus, which handles the accumulation of quantities and areas under and between curves.

In differential calculus, the properties of the natural logarithm simplify the process of finding derivatives, particularly when dealing with exponential functions. Inversely, when we tackle integrals, the natural logarithm often appears when we integrate functions that have the form of 1/x.

Furthermore, the 'change of base' formula from the original exercise showcases how logarithms of any base can be expressed in terms of natural logarithms, making them more accessible for calculus-related problems. To put it succinctly, calculus and logarithms are interwoven, each providing tools that help us understand the language of the universe—mathematics.