Math Problem Statement

ln(|6+t+t^3|)

Solution

The expression given is:

ln(6+t+t3)\ln\left(\left|6 + t + t^3\right|\right)

Here's a breakdown of this expression:

  1. Inside the Absolute Value: The expression 6+t+t36 + t + t^3 could potentially be positive or negative depending on the value of tt. That's why the absolute value function |\cdot| is applied.

  2. Logarithm of the Absolute Value: The natural logarithm function ln\ln is defined only for positive arguments. Thus, the absolute value guarantees that 6+t+t36 + t + t^3 becomes positive, allowing the ln\ln function to be applied.

  3. Domain Analysis: To determine the domain of the function, we must find where 6+t+t3|6 + t + t^3| is greater than zero. This occurs whenever 6+t+t306 + t + t^3 \neq 0.

Simplified Domain of ln(6+t+t3)\ln\left(\left|6 + t + t^3\right|\right)

The domain is all tt except for the roots of 6+t+t36 + t + t^3. Let’s find these roots by solving:

6+t+t3=06 + t + t^3 = 0

This is a cubic equation, which typically requires numerical or algebraic methods to find the roots.

Differentiation of ln(6+t+t3)\ln\left(\left|6 + t + t^3\right|\right)

If you are interested in the derivative of the expression, we apply the chain rule. The derivative of lnu\ln\left|u\right| with respect to uu is given by:

ddtln(6+t+t3)=16+t+t3ddt(6+t+t3)\frac{d}{dt} \ln\left(\left|6 + t + t^3\right|\right) = \frac{1}{6 + t + t^3} \cdot \frac{d}{dt}(6 + t + t^3)

The derivative of 6+t+t36 + t + t^3 is:

ddt(6+t+t3)=1+3t2\frac{d}{dt}(6 + t + t^3) = 1 + 3t^2

Thus, the derivative becomes:

1+3t26+t+t3\frac{1 + 3t^2}{6 + t + t^3}

Questions You Might Consider Next

  1. Would you like a more detailed exploration of the cubic equation 6+t+t3=06 + t + t^3 = 0 and its roots?
  2. Are you interested in the second derivative of this logarithmic function?
  3. Do you want a graphical representation of ln(6+t+t3)\ln\left(\left|6 + t + t^3\right|\right)?
  4. Should we analyze the critical points or inflection points of this function?
  5. Would you like help understanding why the logarithm function requires a positive input?

Tip: When dealing with logarithmic functions involving absolute values, always remember to carefully consider the domain restrictions imposed by the logarithm.

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Math Problem Analysis

Mathematical Concepts

Algebra
Logarithms
Cubic Equations
Differentiation

Formulas

ln|u|, where u = 6 + t + t^3
Derivative of ln|u|: (1/u) * (du/dt)
Cubic equation: 6 + t + t^3 = 0

Theorems

Chain Rule for Differentiation
Properties of Logarithms

Suitable Grade Level

Grades 10-12