Solving Exponential Functions

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What is an exponential function?

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An exponential function is a mathematical function of the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent.

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What does it mean to solve an exponential equation?

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To solve an exponential equation means to find the value of the variable that makes the equation true, often by rewriting both sides of the equation with the same base.

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How do you rewrite a number as a power of another number?

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To rewrite a number as a power of another number, express it in the form of base^exponent, where the base is the number you want to use and the exponent is determined by how many times the base must be multiplied to equal the original number.

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What is the base of an exponential function?

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The base of an exponential function is the constant 'b' in the function f(x) = a * b^x, which determines the growth or decay rate of the function.

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What is the exponent in an exponential function?

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The exponent in an exponential function is the variable 'x' in the function f(x) = a * b^x, which represents the power to which the base is raised.

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How do you solve the equation 36 = 6^x?

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Rewrite 36 as 6^2, so the equation becomes 6^2 = 6^x. Therefore, x = 2.

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What is the solution to the equation 3^3 = 3^(4x + 2)?

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Since the bases are the same, set the exponents equal: 3 = 4x + 2. Solving for x gives x = ¼.

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How do you solve 2^(3n) = 4?

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Rewrite 4 as 2^2, so the equation becomes 2^(3n) = 2^2. Therefore, 3n = 2, and n = 2/3.

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What is the solution to the equation 9^(8-x) = 27^(x-3)?

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Rewrite 9 as 3^2 and 27 as 3^3. The equation becomes (3^2)^(8-x) = (3^3)^(x-3). Set the exponents equal to solve for x, yielding x = 5.

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What is the relationship between exponential functions and logarithms?

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Logarithms are the inverse operations of exponential functions. If b^y = x, then log_b(x) = y.