How to distribute ln into right side of this equation

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SUMMARY

The discussion focuses on the application of the natural logarithm (ln) in differentiation, specifically when dealing with the equation involving the function 976[ (.835)^t - 1] + 176t. It is established that ln cannot be distributed across a sum, as it is not a factor, similar to how cosine and square root functions operate. The chain rule is emphasized as the appropriate method for differentiating ln(f(t)), where f(t) represents the function in question.

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  • Understanding of natural logarithms (ln)
  • Familiarity with differentiation techniques, particularly the chain rule
  • Basic knowledge of functions and their properties
  • Concept of the distributive property in mathematics
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  • Study the chain rule in calculus for differentiating composite functions
  • Explore properties of logarithmic functions, particularly ln and its applications
  • Learn about the differentiation of exponential functions
  • Investigate the limitations of function properties, such as ln, cos, and √
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Students and educators in mathematics, particularly those studying calculus, as well as anyone looking to deepen their understanding of logarithmic differentiation and function properties.

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once i get ln on both sides, I'm then supposed to differentiate both sides...but I won't be able to properly differentiate if ln is not applied.

Does ln get applied to the 976, (-1) and 176t? How?
 
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You can use the chain rule here.

Treat 976[ (.835)^t - 1] + 176t as a function of t. Then your equation becomes ln( f(t) ). And you can use the chain rule to determine the derivative.
 
To answer the question in your title, you can't "distribute" ln into a sum, because ln is not a factor.

Distributive property: a(b + c) = ab + ac

BUT
cos(b + c) ≠ cos(b) + cos(c)
ln(b + c) ≠ ln(b) + ln(c)
√(b + c) ≠ √b + √c

cos, ln, and √ are functions - there is no distributive property for functions.
 

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