Question

A property of logarithms is that \(\log _{a} x=\frac{\log _{b} x}{\log _{b} a},\) for all bases \(a, b>0, \neq 1\) (a) Rewrite this identity when \(b=e\), i.e., using \(\log _{e} x=\) \(\ln x,\) with \(a=10\) (b) Use part \((a)\) to find the derivative of \(y=\log _{10} x\). (c) Use part (b) to find the derivative of \(y-\log _{a} x\), for any \(a>0, \neq 1\).

Short answer

(a) \( \log_{10} x = \frac{\ln x}{\ln 10} \); (b) \( \frac{1}{x \ln 10} \); (c) \( \frac{1}{x \ln a} \) for any \( a > 0, a \neq 1 \).

Step-by-step solution

Rewrite the Log Identity with b=e and a=10

Starting with the identity \( \log_a x = \frac{\log_b x}{\log_b a} \), substitute \( b = e \) and \( a = 10 \). Remember that \( \log_e x \) is written as \( \ln x \). Thus, the identity becomes \( \log_{10} x = \frac{\ln x}{\ln 10} \).

Apply the Chain Rule to Find Derivative of y = log_10 x

Using \( y = \log_{10} x = \frac{\ln x}{\ln 10} \), differentiate both sides with respect to \( x \). The term \( \ln x \) differentiates to \( \frac{1}{x} \), so we have \( \frac{dy}{dx} = \frac{1}{\ln 10} \cdot \frac{d}{dx}(\ln x) = \frac{1}{x \ln 10} \).

Generalize Derivative for y = log_a x Using Change of Base Formula

Using the formula \( \log_a x = \frac{\ln x}{\ln a} \), differentiate with respect to \( x \). Applying the chain rule, \( \frac{d}{dx}(\frac{\ln x}{\ln a}) = \frac{1}{\ln a} \cdot \frac{d}{dx}(\ln x) = \frac{1}{x \ln a} \). This is the derivative of \( y = \log_a x \) for any \( a > 0, a eq 1 \).

Key concepts

Change of Base Formula

When working with logarithms, sometimes you need to change the base of a logarithmic expression in order to simplify or solve equations. The change of base formula helps accomplish this. This formula is expressed as

• \(\log_{a} x = \frac{\log_{b} x}{\log_{b} a}\)

where \(a\) and \(b\) are the bases with \(a, b > 0\) and both \(aeq 1\) and \(beq 1\). This formula is useful when you want to calculate the logarithm with a base other than the one directly available on a calculator or software tool.In the specific from the original step, the base \(b\) is chosen to be \(e\), which is a constant approximately 2.71828 and often arises in mathematical contexts involving growth rates or logarithm transformations. So when we choose \(b=e\), we switch to the natural logarithm, noted as \(\ln x\). For instance, rewriting \(\log_{10} x\) when \(b=e\) gets us:

• \(\log_{10} x = \frac{\ln x}{\ln 10}\)

This allows us to work with natural logarithms, which can simplify differentiation and integration tasks.

Chain Rule in Calculus

The chain rule is an essential tool in calculus used to differentiate complex functions. It is applied when you have a function nested within another function, which is exactly what happens when differentiating logarithms that involve a change of base.Consider the derivative of \(y = \log_{10} x\) using the expression \(y = \frac{\ln x}{\ln 10}\). Here, the outer function is a division by a constant and the inner function is \(\ln x\). Differentiate it using the chain rule:

• The derivative of \(\ln x\) with respect to \(x\) is \(\frac{1}{x}\).
• Since \(\ln 10\) is a constant, \(\frac{1}{\ln 10}\) remains constant in the derivative.

Therefore, the derivative we find is \( \frac{dy}{dx} = \frac{1}{x \ln 10} \). The chain rule here allows us to systematically handle the composed functions step-by-step, ensuring no part of the derivative is missed. This concept becomes even more handy in a broader range of applications beyond just logarithms.

Natural Logarithm

A natural logarithm, denoted by \(\ln\), is a logarithm to the base \(e\), where \(e\) is an irrational and transcendental constant approximately equal to 2.71828. The natural logarithm is widely used in calculus due to its desirable properties.When dealing with derivatives, \(\ln x\) is particularly useful. The derivative of \(\ln x\) with respect to \(x\) is simply \(\frac{1}{x}\). This clean result greatly simplifies calculations involving complex exponential and logarithmic expressions.The natural logarithm is also prolific in various natural phenomena, including growth processes in biology, economics, and many areas where exponential growth is modeled. When differentiating functions that involve any form of logarithm, switching to natural logarithm using the change of base can make the task more straightforward.For example, using \(\ln\) while differentiating \(\log_a x\), leverages the identity:

• \(\log_a x = \frac{\ln x}{\ln a}\)

Thus, we achieve a simple derivative

• \(\frac{dy}{dx} = \frac{1}{x \ln a}\), making use of the properties of \(\ln\).

Emphasis on understanding and utilizing \(\ln\) can be pivotal for tackling a variety of problems across disciplines.