How do you solve the equation $$27^{(-x+2)} = \frac{1}{243}$$ ?

To solve this equation, first express both sides with the same base. Since $$27 = 3^3$$ and $$243 = 3^5$$ , rewrite the equation as $$(3^3)^{(-x+2)} = (3^{-5})$$ . This simplifies to $$3^{-3x + 6} = 3^{-5}$$ . Set the exponents equal: $$-3x + 6 = -5$$ . Solving gives $$x = \frac{11}{3}$$ .

What is the change of base formula for logarithms?

The change of base formula allows you to convert a logarithm to a different base: $$\text{log}_b(a) = \frac{\text{log}_k(a)}{\text{log}_k(b)}$$ , where $$k$$ is any positive number.

What is the definition of a power function?

A power function is a function of the form $$f(x) = kx^n$$ , where $$k$$ is a constant and $$n$$ is a real number.

What is the relationship between exponential growth and decay?

Exponential growth occurs when a quantity increases at a rate proportional to its current value, while exponential decay occurs when a quantity decreases at a rate proportional to its current value.

What is the formula for exponential growth?

The formula for exponential growth is $$N(t) = N_0 e^{rt}$$ , where $$N(t)$$ is the amount at time $$t$$ , $$N_0$$ is the initial amount, $$r$$ is the growth rate, and $$e$$ is the base of the natural logarithm.

What is the formula for exponential decay?

The formula for exponential decay is $$N(t) = N_0 e^{-rt}$$ , where $$N(t)$$ is the amount at time $$t$$ , $$N_0$$ is the initial amount, $$r$$ is the decay rate, and $$e$$ is the base of the natural logarithm.

What is the significance of the number $$e$$ in exponential functions?

The number $$e$$ , approximately equal to 2.71828, is the base of natural logarithms and is significant in continuous growth or decay processes.

What is a common application of exponential functions in real life?

Exponential functions are commonly used to model population growth, radioactive decay, and interest calculations in finance.

HW Flashcard 10.5 (ALL) Flashcards

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14 questions

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1.

FLASHCARD QUESTION

Front

What is the definition of an exponential equation?

Back

2.

FLASHCARD QUESTION

Front

Back

3.

FLASHCARD QUESTION

Front

What is the base of an exponential function?

Back

4.

FLASHCARD QUESTION

Front

What is the inverse of an exponential function?

Back

5.

FLASHCARD QUESTION

Front

Back

This property is known as the Product of Powers Property, which states that when multiplying two exponential expressions with the same base, you add the exponents.

6.

FLASHCARD QUESTION

Front

Back

This property is known as the Quotient of Powers Property, which states that when dividing two exponential expressions with the same base, you subtract the exponents.

7.

FLASHCARD QUESTION

Front

What is the definition of a logarithm?

Back

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