5 Questions
Express how a horizontal dilation of the graph of an exponential function is related to a change in the base of the function.
A horizontal dilation of the graph of an exponential function is equivalent to a change of the base of the function because $f(x)= b^{kx}$ can be expressed as $f(x) = (b^{k})^{x}$ where $b^{k}$ is constant.
How can you construct a representation of an exponential function using the Natural Base, e, to reveal properties of the graph or contextual scenarios?
To construct a representation of an exponential function using the Natural Base, e, one can express it in an equivalent form using properties of exponents, function composition, and properties of logarithms.
Explain how the perspective of an exponential model can be expressed in any base, including the natural base, using properties of exponents and function composition.
The perspective of an exponential model can be expressed in any base, including the natural base, by leveraging properties of exponents and function composition to represent the model equivalently.
Why is the expression $f(x) = b^{kx}$ equivalent to $f(x) = (b^{k})^{x}$ in terms of the constant $b^{k}$?
The expression $f(x) = b^{kx}$ is equivalent to $f(x) = (b^{k})^{x}$ because $b^{k}$ is a constant that remains unchanged under different forms of the exponential function.
Discuss the significance of using the Natural Base, e, in expressing exponential functions and revealing graph properties.
Using the Natural Base, e, to express exponential functions unveils key graph properties and facilitates a deeper understanding of exponential growth and decay phenomena.
Study Notes
Exponential Functions in Natural Base (e)
- Exponential functions can be represented using the natural base (e) to reveal properties of the graph or contextual scenarios.
- Any exponential model can be expressed in any base, including the natural base, using properties of exponents and/or function composition and properties of logarithms.
Horizontal Dilation and Change of Base
- A horizontal dilation of the graph of an exponential function is equivalent to a change of the base of the function.
- The function f(x) = b^kx can be expressed as f(x) = (b^k)^x, where b^k is a constant.
- This equivalence reveals the relationship between horizontal dilation and change of base in exponential functions.
Test your understanding of constructing a representation of an exponential function using the Natural Base e to reveal properties of the graph or contextual scenarios. Explore equivalent forms of exponential functions and apply properties of exponents, function composition, and logarithms.
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