Exponential and Logarithmic Equations

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What is the definition of an exponential equation?

Back

An exponential equation is an equation in which a variable appears in the exponent, typically in the form of a^x = b, where a is a positive constant, b is a positive number, and x is the variable.

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What is the definition of a logarithmic equation?

Back

A logarithmic equation is an equation that involves a logarithm of a variable, typically in the form of log_a(x) = b, where a is the base, x is the argument, and b is the result.

Front

What is the relationship between exponential and logarithmic functions?

Back

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

Front

How do you solve an exponential equation of the form e^(kx) = C?

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To solve e^(kx) = C, take the natural logarithm of both sides: kx = ln(C), then solve for x: x = ln(C)/k.

Front

What is the change of base formula for logarithms?

Back

The change of base formula states that log_a(b) = log_c(b) / log_c(a) for any positive c.

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What is the product property of logarithms?

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The product property states that log_a(xy) = log_a(x) + log_a(y).

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What is the quotient property of logarithms?

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The quotient property states that log_a(x/y) = log_a(x) - log_a(y).

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What is the power property of logarithms?

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The power property states that log_a(x^b) = b * log_a(x).

Front

How do you solve logarithmic equations?

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To solve logarithmic equations, isolate the logarithm on one side, convert to exponential form, and then solve for the variable.

Front

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 2.718.