Solving Exponential and Logarithmic Equations

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What is an exponential equation?

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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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What is a logarithmic equation?

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A logarithmic equation is an equation that involves a logarithm of a variable. For example, log_b(x) = y means that b^y = x.

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What is the change of base formula for logarithms?

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The change of base formula states that log_b(a) = log_k(a) / log_k(b) for any positive k, where k is not equal to 1.

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How do you combine logarithms with the same base?

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You can combine logarithms with the same base using the properties: log_b(m) + log_b(n) = log_b(m*n) and log_b(m) - log_b(n) = log_b(m/n).

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What is the product property of logarithms?

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The product property of logarithms states that the logarithm of a product is the sum of the logarithms: log_b(m*n) = log_b(m) + log_b(n).

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What is the quotient property of logarithms?

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The quotient property of logarithms states that the logarithm of a quotient is the difference of the logarithms: log_b(m/n) = log_b(m) - log_b(n).

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What is the power property of logarithms?

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The power property of logarithms states that the logarithm of a number raised to a power is the power times the logarithm of the number: log_b(m^k) = k * log_b(m).

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How do you solve an exponential equation?

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To solve an exponential equation, you can take the logarithm of both sides, use properties of logarithms to simplify, and then solve for the variable.

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How do you solve a logarithmic equation?

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To solve a logarithmic equation, you can exponentiate both sides to eliminate the logarithm, then solve the resulting equation for the variable.

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What is the relationship between exponential and logarithmic functions?

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Exponential functions and logarithmic functions are inverses of each other. If y = b^x, then x = log_b(y).