Why the answers are different if theyre the same

  • Context: Undergrad 
  • Thread starter Thread starter alba_ei
  • Start date Start date
Click For Summary
SUMMARY

The discussion centers on the differences in integral results for the expressions involving logarithms, specifically the integral of \(2 \ln(x)\) versus the integral of \(\ln(x)^2\). It clarifies that if \(\ln(x)^2\) refers to \((\ln(x))^2\), the two functions are not equivalent, leading to different integral results. Conversely, if \(\ln(x)^2\) is interpreted as \(\ln(x^2)\), then the integrals yield the same answer due to the logarithmic identity \(\ln(x^2) = 2\ln(x)\). This highlights the importance of precise notation in calculus.

PREREQUISITES
  • Understanding of integral calculus
  • Familiarity with logarithmic properties
  • Knowledge of function notation and interpretation
  • Basic skills in mathematical notation
NEXT STEPS
  • Study the properties of logarithms in depth, focusing on identities like \(\log_a(b^c) = c\log_a(b)\)
  • Practice solving integrals involving logarithmic functions
  • Explore the differences between function notation and its implications in calculus
  • Learn about common pitfalls in interpreting mathematical expressions
USEFUL FOR

Students of calculus, mathematics educators, and anyone looking to deepen their understanding of logarithmic functions and their integrals.

alba_ei
Messages
38
Reaction score
1
why the answers are different?
-integral of 2 ln(x)
-integral of ln(x)^2
 
Physics news on Phys.org
by ln(x)^2 do you mean (ln(x))2 or ln(x2)? If you meant the second one then your answers should not be different as they are exactly the same, however, if you meant the first then the two functions are not equal and you should not expect equivalent answers.
 
[tex]\ln(x^2)=2\ln(x)[/tex]. More generally, it is a property of logarithms that [tex]\log_a(b^c)=c\log_a(b)[/tex].
 

Similar threads

Undergrad Why Are There Different Forms of the Integration Formula for Cosecant?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Having trouble understanding a seemingly simple u-substitution
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Did Math DF's integral calculator make a glaring mistake?
  • · Replies 8 ·
Replies
8
Views
4K
Undergrad Integrating 1/x with units (logarithm)
  • · Replies 15 ·
Replies
15
Views
4K
High School Integral of cosecant function: understanding different approaches
  • · Replies 14 ·
Replies
14
Views
4K
High School Are Both Answers Correct for Trigonometric Substitution Integral?
  • · Replies 5 ·
Replies
5
Views
3K
High School Calculating shaded area: I'm getting discrepancies between methods
  • · Replies 3 ·
Replies
3
Views
3K
MHB Why Doesn't the Intermediate Value Theorem Apply to ln(x^2 + 2) on [−2, 2]?
  • · Replies 5 ·
Replies
5
Views
2K
High School Why is the definite integral of 1/x from -1 to 1 undefined?
  • · Replies 4 ·
Replies
4
Views
7K
High School Learn 0^0 w/ Young Aussie of the Year: Calculus & Real Analysis Explained
  • · Replies 14 ·
Replies
14
Views
2K