Precalc - 3.4(1) Exponential Eqs Using Logs

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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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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 the inverse function of the exponential function.

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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, if you have e^x = a, you can take ln(e^x) = ln(a), which simplifies to x = ln(a).

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What does it mean if an exponential equation has no solution?

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An exponential equation has no solution if the equation results in a contradiction, such as a negative number on the left side when the right side is positive.

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How do you isolate the variable in an exponential equation?

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To isolate the variable in an exponential equation, you can use logarithms to bring the exponent down. For example, in the equation 5^x = 25, you can take log base 5 of both sides to find x.

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What is the base of a logarithm?

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The base of a logarithm is the number that is raised to a power to obtain a given number. For example, in log_b(a) = c, b is the base.

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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).

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How do you convert an exponential equation to logarithmic form?

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To convert an exponential equation to logarithmic form, rewrite it as log_b(a) = c, where b^c = a. For example, if 2^3 = 8, it can be written as log_2(8) = 3.

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What is the value of ln(1)?

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The value of ln(1) is 0, because e^0 = 1.

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What is the value of e^0?

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The value of e^0 is 1, as any non-zero number raised to the power of 0 equals 1.