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(2^x^2-4^x)(log3(x+25)-3)

4 min read Jun 16, 2024
(2^x^2-4^x)(log3(x+25)-3)

Analyzing the Expression (2^(x^2-4^x))(log3(x+25)-3)

This expression combines exponential and logarithmic functions, making it interesting to analyze. Let's break it down and explore its key features:

Understanding the Components

  • Exponential Part: (2^(x^2-4^x))
    • The base is 2, a constant value.
    • The exponent (x^2-4^x) is a polynomial function, involving both x^2 and 4^x. This part introduces a complex relationship between the input (x) and the output of the exponential function.
  • Logarithmic Part: (log3(x+25)-3)
    • The base is 3, a constant value.
    • The argument of the logarithm is (x+25), a linear function.
    • The constant -3 shifts the entire logarithmic function downwards by 3 units.

Key Observations

  • Domain: The expression is defined for values of x where:

    • (x+25) > 0: This ensures the argument of the logarithm is positive, as logarithms are only defined for positive values.
    • (x^2 - 4^x) is defined: The exponent can be any real number, so this condition is always met.
    • Overall: The domain of the expression is x > -25.

  • Behavior:

    • Exponential Growth: The exponential part (2^(x^2-4^x)) will experience rapid growth as x increases.
    • Logarithmic Growth: The logarithmic part (log3(x+25)-3) will experience growth, but at a slower rate than the exponential part.
    • Interaction: The interaction between these two components is complex, leading to interesting behavior as x varies.

Solving for Zeroes

Finding the zeroes of this expression requires solving the equation:

(2^(x^2-4^x))(log3(x+25)-3) = 0

This means either:

  • 2^(x^2-4^x) = 0: This equation has no solutions since exponential functions are always positive.
  • log3(x+25)-3 = 0: Solving this, we get:
    • log3(x+25) = 3
    • x+25 = 3^3 = 27
    • x = 2.

Key Points to Consider

  • Graphical Representation: The expression's graph will exhibit complex shapes due to the combined effects of the exponential and logarithmic components.
  • Numerical Methods: Analyzing the expression's behavior for specific values of x can be done using numerical methods or software.
  • Applications: This type of expression may arise in modeling various phenomena involving exponential growth, logarithmic scaling, or their interactions.

By understanding the individual components and their interactions, we gain valuable insights into the behavior and properties of this expression.

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