Solving Exponential Equations

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What is an exponential equation?

Back

An exponential equation is an equation in which a variable appears in the exponent. For example, in the equation 2^x = 8, x is the variable in the exponent.

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How do you solve the exponential equation 5^{-3x} = 125?

Back

To solve 5^{-3x} = 125, rewrite 125 as 5^3. This gives you 5^{-3x} = 5^3. Since the bases are the same, set the exponents equal: -3x = 3, leading to x = -1.

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What is the first step in solving the equation 4^(p+2) = 64?

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The first step is to express 64 as a power of 4. Since 64 = 4^3, you can rewrite the equation as 4^(p+2) = 4^3 and then set the exponents equal.

Front

How do you convert the equation 10^(2x+4) = 100000 into a simpler form?

Back

Rewrite 100000 as a power of 10. Since 100000 = 10^5, the equation becomes 10^(2x+4) = 10^5, allowing you to set the exponents equal.

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What is the solution to the equation 5^{-3x - 1} = 25?

Back

Rewrite 25 as 5^2. The equation becomes 5^{-3x - 1} = 5^2. Set the exponents equal: -3x - 1 = 2, leading to x = -1.

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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 that is raised to a power. In the function f(x) = a^x, 'a' is the base.

Tags

CCSS.HSF-IF.C.8B

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How do you solve for x in the equation 8 = 2^{5x+7}?

Back

Rewrite 8 as a power of 2: 8 = 2^3. The equation becomes 2^3 = 2^{5x+7}. Set the exponents equal: 3 = 5x + 7, leading to x = -4/5.

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What is the property of exponents that allows you to set exponents equal when the bases are the same?

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This property is known as the equality of exponents property, which states that if a^m = a^n (where a > 0 and a ≠ 1), then m = n.

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What is the logarithmic form of the exponential equation 2^x = 16?

Back

The logarithmic form is x = log_2(16). Since 16 = 2^4, this simplifies to x = 4.

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CCSS.HSF.BF.B.5

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How can you check your solution for an exponential equation?

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You can check your solution by substituting the value back into the original equation to see if both sides are equal.