Textbook Question
In Exercises 61–64, give the equation of each exponential function whose graph is shown.
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6. Exponential & Logarithmic Functions
The Number e
3:30 minutesProblem 91
Textbook Question
Solve each equation. See Examples 4–6. (1/e)^-x = (1/e^2)^(x+1)
Verified step by step guidance1
Start by recognizing that both sides of the equation have the same base, which is \( \frac{1}{e} \).
Rewrite the equation using the property of exponents: \( \left( \frac{1}{e} \right)^{-x} = \left( \frac{1}{e^2} \right)^{x+1} \).
Express \( \frac{1}{e^2} \) as \( \left( \frac{1}{e} \right)^2 \) to have the same base on both sides: \( \left( \frac{1}{e} \right)^{-x} = \left( \left( \frac{1}{e} \right)^2 \right)^{x+1} \).
Apply the power of a power property \((a^m)^n = a^{m \cdot n}\) to simplify the right side: \( \left( \frac{1}{e} \right)^{-x} = \left( \frac{1}{e} \right)^{2(x+1)} \).
Since the bases are the same, set the exponents equal to each other: \(-x = 2(x+1)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Functions
Exponential functions are mathematical expressions in the form of f(x) = a * b^x, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. These functions exhibit rapid growth or decay and are fundamental in solving equations involving exponents, as they allow us to manipulate and equate powers effectively.
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Properties of Exponents
The properties of exponents are rules that govern how to manipulate expressions involving powers. Key properties include the product of powers (a^m * a^n = a^(m+n)), the quotient of powers (a^m / a^n = a^(m-n)), and the power of a power ( (a^m)^n = a^(m*n)). Understanding these properties is essential for simplifying and solving exponential equations.
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Logarithms
Logarithms are the inverse operations of exponentiation, allowing us to solve for exponents in equations. The logarithm log_b(a) answers the question: 'To what exponent must the base b be raised to produce a?' This concept is crucial when dealing with equations where the variable is in the exponent, as it enables us to isolate the variable and find its value.
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