Solve Exponential Equations using Logarithms

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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 the logarithm of a number?

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The logarithm of a number is the exponent to which a base must be raised to produce that number. For example, log₂(8) = 3 because 2³ = 8.

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How do you solve an exponential equation using logarithms?

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To solve an exponential equation using logarithms, take the logarithm of both sides of the equation. For example, to solve 2^x = 8, take log₂ of both sides: x = log₂(8).

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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. This allows you to calculate logarithms with different bases.

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What is the natural logarithm?

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The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately equal to 2.71828. It is used in many areas of mathematics and science.

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What is the common logarithm?

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The common logarithm is the logarithm to the base 10, denoted as log(x). It is often used in scientific calculations.

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What is the property of logarithms that states log_b(xy) = ?

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log_b(x) + log_b(y). This property allows you to combine the logarithms of products.

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What is the property of logarithms that states log_b(x/y) = ?

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log_b(x) - log_b(y). This property allows you to separate the logarithm of a quotient.

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What is the property of logarithms that states log_b(xⁿ) = ?

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n * log_b(x). This property allows you to bring the exponent in front of the logarithm.

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How do you solve the equation 7ⁿ⁺¹⁰ - 8 = 6?

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First, isolate the exponential term: 7ⁿ⁺¹⁰ = 14. Then take the logarithm of both sides: n + 10 = log₇(14). Finally, solve for n: n = log₇(14) - 10.