How do you solve the equation $$64^x = 8^{3x+1}$$ ?

Convert both sides to the same base: $$64 = 8^2$$ , so $$(8^2)^x = 8^{3x+1}$$ . This simplifies to $$8^{2x} = 8^{3x+1}$$ . Setting the exponents equal gives $$2x = 3x + 1$$ , leading to $$x = -1$$ .

How do you solve the equation $$32^{x+6} \times \frac{1}{2} = 8^{x-1}$$ ?

Convert to the same base: $$32 = 2^5$$ and $$8 = 2^3$$ . Rewrite the equation as $$(2^5)^{x+6} \times 2^{-1} = (2^3)^{x-1}$$ . This simplifies to $$2^{5x + 30 - 1} = 2^{3x - 3}$$ . Setting exponents equal gives $$5x + 29 = 3x - 3$$ , leading to $$x = -16$$ .

What is the general form of an exponential decay function?

The general form of an exponential decay function is $$f(x) = a \times b^x$$ , where $$a > 0$$ and $$0

How do you determine if the function $$f(x) = \frac{1}{6} \times 4^x$$ is growth or decay?

Since the base $$4 > 1$$ , the function exhibits growth.

How do you solve the equation $$5^{3x-1} \times 5^{2x-5} = 5^{x+6}$$ ?

Combine the left side: $$5^{(3x-1) + (2x-5)} = 5^{x+6}$$ . This simplifies to $$5^{5x - 6} = 5^{x + 6}$$ . Setting exponents equal gives $$5x - 6 = x + 6$$ , leading to $$x = 3$$ .

What is the property of exponents that allows you to combine terms like $$a^m \times a^n = a^{m+n}$$ ?

This property states that when multiplying like bases, you can add the exponents.

What is the property of exponents that allows you to simplify terms like $$\frac{a^m}{a^n} = a^{m-n}$$ ?

This property states that when dividing like bases, you can subtract the exponents.

What is the inverse of an exponential function?

The inverse of an exponential function is a logarithmic function, expressed as $$f^{-1}(x) = \text{log}_b(x)$$ , where $$b$$ is the base of the exponential function.

What is the significance of the y-intercept in an exponential function?

The y-intercept of an exponential function $$f(x) = a \times b^x$$ is the value of $$f(0) = a$$ , which indicates the initial value of the function.

How can you graph an exponential function?

To graph an exponential function, plot points for various values of $$x$$ , noting the y-intercept and the behavior as $$x$$ approaches positive and negative infinity.

Exponential functions and Equations Flashcards

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16 questions

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1.

FLASHCARD QUESTION

Front

What is an exponential function?

Back

2.

FLASHCARD QUESTION

Front

What is the base of an exponential function?

Back

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