5. e^(2t)-3e^t = 0

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Rewrite the equation \(e^{2t} - 3e^t = 0\) by recognizing that \(e^{2t}\) can be expressed as \((e^t)^2\). This allows us to treat the equation like a quadratic in terms of \(e^t\).

Let \(x = e^t\). Substitute this into the equation to get \(x^2 - 3x = 0\).

Factor the quadratic equation: \(x(x - 3) = 0\). This gives two possible solutions for \(x\): \(x = 0\) or \(x = 3\).

Recall that \(x = e^t\), and since \(e^t\) is never zero for any real \(t\), discard \(x = 0\). Focus on \(e^t = 3\).

Solve for \(t\) by taking the natural logarithm of both sides: \(t = \ln(3)\). This gives the solution for \(t\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponential Functions

Exponential functions involve variables in the exponent, such as e^t. Understanding their properties, like how e^(a+b) = e^a * e^b, helps simplify and solve equations involving exponentials.

Substitution Method

The substitution method replaces a complex expression with a simpler variable to transform the equation into a more familiar form, such as turning e^t into x, making it easier to solve.

Solving Quadratic Equations

After substitution, the equation often becomes quadratic. Knowing how to solve quadratic equations using factoring, the quadratic formula, or completing the square is essential to find the variable's values.

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