Solving f'(x) of log10 x: A Step-by-Step Guide

  • Thread starter nobahar
  • Start date
In summary, the conversation discusses finding the derivative of log base 10 x, with one person initially arriving at 1/x * log10e and the other at 1/(x ln 10). The discrepancy is explained by the change of base property of logarithms, where log base 10 x can be written as ln x over ln 10. Both answers are confirmed to be correct. The conversation ends with a polite exchange between the two individuals.
  • #1
nobahar
497
2

Homework Statement


f'(x) of log10 x


Homework Equations


It's quite a nightmare to write on here if your not particularly fast with the code thing!

The Attempt at a Solution


I eventually reached 1/x * log10e
However, the answer should be 1/(x ln 10).
Is there a stage from what I obtained that leads to the answer?
If not, then there's something incorrect 'further up' the calculations and I'll go back and have a look... Just would appreciate some input as to whether or not I've arrived at a correct stage.
Thanks.
 
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  • #2
Your answer is correct. The discrepancies between the two answers can be easily resolved with the change of base property of logarithms.

Ex: (log_a (b)) = 1/(log_b (a)).
 
  • #3
For any numbers, a, b, x,
[tex]log_b(x)= \frac{log_a(x)}{log_a(b)}[/tex]

In particular,
[tex]log_{10}(x)= \frac{ln(x)}{ln(10)}[/itex]

Therefore
[tex]\frac{dlog_{10}(x)}{dx}= \frac{d ln(x)}{dx}\frac{1}{ln(10)}[/tex]
[tex]= \frac{1}{x}\frac{1}{ln(10)}= \frac{1}{x ln(10)}[/tex]

BUT, by that same initial formula,
[tex]log_{10}(e)= \frac{ln(e)}{ln(10}= \frac{1}{ln(10)}[/tex]

so your answer is also perfectly correct!

In general
[tex]log_a(b)= \frac{1}{log_b(a)}[/tex]
 
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  • #4
Thanks!
I arrived at that just now. Thought I'd log into update my post and ask to see if the method was correct, but it's been confirmed! Thanks for the quick responses (and the detail)!
:smile: Speak soon (hopefully not too soon, otherwise it means I'm stuck).
 
  • #5
Speak what soon? You said that you have already confirmed that what you had was correct and two people have already agreed!
 
  • #6
No! I was just being polite!:smile:
You also arrived at the answer through a far quicker method than I used...
 

1. What is the purpose of solving f'(x) of log10 x?

The purpose of solving f'(x) of log10 x is to find the derivative of the logarithmic function log10 x. This can be useful in many mathematical and scientific applications, such as finding rates of change or slopes of curves.

2. How do I solve f'(x) of log10 x?

To solve f'(x) of log10 x, you can use the power rule for logarithms, which states that the derivative of log base a of x is equal to 1 divided by x times the natural logarithm of a. In this case, a is equal to 10, so the derivative of log10 x is 1 divided by x times the natural logarithm of 10.

3. Can I use the chain rule to solve f'(x) of log10 x?

Yes, you can use the chain rule to solve f'(x) of log10 x. The chain rule states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function. In this case, the outer function is log10 x and the inner function is x. So, the derivative of log10 x is equal to 1 divided by x times the derivative of x, which is 1.

4. Are there any special cases to consider when solving f'(x) of log10 x?

Yes, there is one special case to consider when solving f'(x) of log10 x. When x is equal to 0, the derivative of log10 x is undefined. This is because the logarithmic function is not defined for x values less than or equal to 0. Therefore, when solving f'(x) of log10 x, be sure to exclude x = 0 from your solution.

5. How can I check if I've solved f'(x) of log10 x correctly?

To check if you've solved f'(x) of log10 x correctly, you can use the quotient rule for derivatives. The quotient rule states that the derivative of f(x) divided by g(x) is equal to the derivative of f(x) times g(x) minus f(x) times the derivative of g(x), all divided by g(x) squared. In this case, f(x) is equal to log10 x and g(x) is equal to x. If you apply the quotient rule and get a result of 1, then you have correctly solved f'(x) of log10 x.

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