How do you solve log in differentiation?

The process of differentiating y=f(x) with logarithmic differentiation is simple. Take the natural log of both sides, then differentiate both sides with respect to x. Solve for dydx and write y in terms of x and you are finished . Consider the function (f(x))(g(x)), for any (non-constant) functions f and g.

How to differentiate log x?

The derivative of log x (base 10) is 1/(x ln 10) . If the log has a base "a", then its derivative is 1/(x ln a).

What is the derivative of log e?

Differentiation of Log e As observed in the previous sections, log e is a constant value and the derivative of a constant term is always equal to zero. Therefore, the derivative of log e is equal to 0 and we can write it as d(log e)/dx = 0 .

What is the derivative of logx base a?

Derivative of loga(x) is 1xln(a) . Here “ln” is the derivative of “log”. “ln” is called the natural logarithm or it is a logarithm with base 'e', i.e. ln=loge.

What is the derivative formula for logs?

What are the Formulas for Derivative of Log Function? (d / dx) ln x = 1 / x where, x >0. (d / dx) log a x = 1 / (x ln a) where, a ≠ 1. (d / dx) ln f(x) = f'(x) / f(x) Jun 25, 2024

How to find log x value?

Answer: The log x value refers to the logarithm of x, which is the exponent to which a base, typically 10 or e, must be raised to produce the number x . For example, if you're calculating the base-10 logarithm of 100, denoted as log(100), the answer is 2 because 10 2 = 100.

What is the formula for differentiation?

What Are Differentiation Formulas? The differentiation formula is used to find the derivative or rate of change of a function. if y = f(x), then the derivative dy/dx = f'(x) = limΔx→0f(x+Δx)−f(x)Δx lim Δ x → 0 ⁡

How do you differentiate 2 log x?

d (2logx)/dx let, u=2, v=logx. Then, d(uv)/dx =udv/dx +vdu/dx. d(2logx)/dx =2d (logx)/dx +logx d(2)/dx . d(2logx)/dx =2/x.

How to calculate log?

logarithm, the exponent or power to which a base must be raised to yield a given number. Expressed mathematically, x is the logarithm of n to the base b if b x = n, in which case one writes x = log b n . For example, 2 3 = 8; therefore, 3 is the logarithm of 8 to base 2, or 3 = log 2 8.

Why use logarithmic differentiation?

While logarithmic differentiation can help us with algebraically tricky questions , this technique's real power is when we are given an expression where one variable is raised to another variable — and the normal rules for derivatives don't apply.

How to convert log into ln?

The relationship between ln x and log x is: ln x = 2.303 log x Why 2.303? Let's use x = 10 and find out for ourselves.

The ln derivative rule says "the derivative of ln x is 1/x". It is mathematically written as follows: d/dx (ln x) = 1/x (or)

Are log and ln the same?

The difference between log and ln is that log is defined for base 10 and ln is denoted for base e . For example, log of base 2 is represented as log 2 and log of base e, i.e. log e = ln (natural log).

How do you differentiate log 7x?

The derivative of log7(x) log 7 ( x ) with respect to x is 1xln(7) 1 x ln ( 7 ) .

What is the value of log 10 in differentiation?

As log of 10 to the base e is a constant value ,so its differentiation must be zero . Log10 with base e = ln10 = 2.3025.

Problem 34 Differentiate. $$G(x)=\log _{9... [FREE SOLUTION] (2024)

Get started for free

Get started for free Log out

Chapter 3: Problem 34

Differentiate. $$G(x)=\log _{9} x \cdot\left(4^{x}\right)^{6}$$

Short Answer

Expert verified

$ G'(x) = \frac{2^{12x}}{x \ln 9} + 12 \log_{9} x \cdot 2^{12x} \cdot \ln 2 $.

Step by step solution

Understand the Function

The function given is $ G(x) = \log_{9} x \cdot(4^x)^{6} $. This function is a product of two functions: $ \log_{9} x $ and $ (4^x)^{6} $. We'll use the product rule to differentiate this function.

Apply the Product Rule

The product rule states: $ (uv)' = u'v + uv' $. Here, let $ u = \log_{9} x $ and $ v = (4^x)^{6} $. First, find the derivatives of $ u $ and $ v $.

Find the Derivative of $ u = \log_{9} x $

We know the derivative of $ \log_{a} x $ is $ \frac{1}{x \ln a} $. Hence, $ u' = \frac{1}{x \ln 9} $.

Simplify $ v $

Rewrite $ v $ to a simpler form to differentiate. $ v = (4^x)^{6} = 4^{6x} = (2^2)^{6x} = 2^{12x} $.

Find the Derivative of $ v = 2^{12x} $

The derivative of $ a^{f(x)} $ is given by $ a^{f(x)} \cdot f'(x) \cdot \ln a $. Here $ f(x) = 12x $ and $ f'(x) = 12 $. Thus, $ v' = 2^{12x} \cdot 12 \cdot \ln 2 $.

Combine Using Product Rule

Apply the product rule: $ G'(x) = u'v + uv' $. Substituting the derivatives found: $ G'(x) = \frac{1}{x \ln 9} \cdot 2^{12x} + \log_{9} x \cdot 2^{12x} \cdot 12 \cdot \ln 2 $.

Simplify the Expression

Combine like terms to simplify: $ G'(x) = \frac{2^{12x}}{x \ln 9} + (12 \log_{9} x \cdot 2^{12x} \cdot \ln 2) $. Finally, rearrange terms for a simplified answer.

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Product Rule

The product rule is a fundamental tool in calculus used to find the derivative of the product of two functions. If you have two differentiable functions, say $ u(x) $ and $ v(x) $, their product is denoted $ u(x) \times v(x) $. The product rule states that the derivative of this product, denoted as $ (uv)' $, is given by:

\[ (uv)' = u'v + uv' \]

Here's how it works in simple terms:

Take the derivative of the first function $ u $, while keeping the second function $ v $ as it is.
Add the product of the first function $ u $ with the derivative of the second function $ v $.

For our exercise, we have $ u = \log_{9} x $ and $ v = (4^x)^6 $. We can differentiate each part separately and then apply the product rule:

Find $ u' $, the derivative of $ \log_{9} x $. This is $ \frac{1}{x \ln 9} $.
Find $ v' $, the derivative of $ (4^x)^6 $ simplified to $ 2^{12x} $. This is $ 2^{12x} \cdot 12 \cdot \ln 2 $.

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions of the form $ y = f(x)^{g(x)} $. It's particularly useful when the function is too complex to differentiate using standard rules. The steps to logarithmic differentiation are:

Take the natural logarithm on both sides of the equation: $ \ln y = \ln (f(x)^{g(x)}) $.
Simplify the right side using logarithm properties: $ \ln y = g(x) \cdot \ln f(x) $.
Differentiate both sides with respect to $ x $: $ \frac{d}{dx} ( \ln y ) = \frac{d}{dx} ( g(x) \cdot \ln f(x) ) $.
Solve for $ y' $: $ y' = y \cdot (\text{result from step 3}) $.

In the context of our given function $ G(x) = \log_{9} x \cdot (4^x)^6 $, we applied logarithm properties to simplify the function before taking derivatives. Differentiating $ u = \log_{9} x $ was straightforward, but differentiating $ v = (4^x)^6 $ required rewrite in simpler forms before using the power and exponential rules.

Exponential Function Differentiation

Exponential functions are those of the form $ a^{g(x)} $, where $ a $ is a positive constant. The differentiation process involves:

Identifying the base $ a $ and the exponent function $ g(x) $.
Using the rule $ \frac{d}{dx} (a^{g(x)}) = a^{g(x)} \cdot g'(x) \cdot \ln a $ to find the derivative.

For example, in our problem, we simplified $ (4^x)^6 $ to $ 2^{12x} $. Then we used the exponential function rule:
\[ \frac{d}{dx} (2^{12x}) = 2^{12x} \cdot 12 \cdot \ln 2 \]
This gave us the derivative of the exponential component of our function. Combining this with the product rule, we were able to fully differentiate the original function $ G(x) $.
Breaking down complex functions using these methods allows us to tackle differentiation in a structured and understandable way.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

$Problem 34 Differentiate. $$G(x)=\log _{9... [FREE SOLUTION] (3)$

Most popular questions from this chapter

The demand for a new computer game can be modeled by $p(x)=53.5-8 \ln x$ where$p(x)$ is the price consumers will pay, in dollars, and $x$ is the number ofgames sold, in thousands. Recall that total revenue is given by $R(x)=x \cdotp(x)$ a) Find $R(x)$ b) Find the marginal revenue, $R^{\prime}(x)$ c) Is there any price at which revenue will be maximized? Why or why not?Intensity of sound. The intensity of a sound is given by $I=I_{0} 10^{0.1 L},$ where $L$ is the loudness of the sound as measured in decibels and $I_{0}$ isthe minimum intensity detectable by the human ear. a) Find $I$, in terms of $I_{0},$ for the loudness of a power mower, which is100 decibels. b) Find $I,$ in terms of $I_{0},$ for the loudness of a just audible sound,which is 10 decibels. c) Compare your answers to parts (a) and (b). d) Find the rate of change $d I / d L$ e) Interpret the meaning of $d I / d L$Differentiate. $$f(x)=15^{x}$$We have now studied models for linear, quadratic, exponential, and logisticgrowth. In the real world, understanding which is the most appropriate type ofmodel for a given situation is an important skill. Identify the mostappropriate type of model and explain why you chose that model. List anyrestrictions you would place on the domain of the function. The drop and rise of a lake's water level during and after a droughtTo what exponential growth rate per hour does a growth rate of $100 \%$ perday correspond?

See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free

This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept

Privacy & Cookies Policy

Privacy Overview

This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience.

Necessary

Always Enabled

Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information.

Non-necessary

Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. It is mandatory to procure user consent prior to running these cookies on your website.

SAVE & ACCEPT

Problem 34 Differentiate. $$G(x)=\log _{9... [FREE SOLUTION] (2024)

Short Answer

Step by step solution

Understand the Function

Apply the Product Rule

Find the Derivative of \( u = \log_{9} x \)

Simplify \( v \)

Find the Derivative of \( v = 2^{12x} \)

Combine Using Product Rule

Simplify the Expression

Key Concepts

Product Rule

Logarithmic Differentiation

Exponential Function Differentiation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Decision Maths

Statistics

Pure Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Privacy Overview

FAQs

How do you solve log in differentiation? ›

How to convert log into ln? ›