Problem 122 - Elementary and Intermediate Algebra

Question

If \(e^{5 t}=k^{t},\) find \(k\)

Step-by-Step Solution

Answer

The value of \( k \) is \( e^5 \).

Step 1: Set Up the Equation

We start with the given equation: \( e^{5t} = k^t \). Both sides of this equation are equal so we can manipulate them to isolate \( k \).

Step 2: Apply the Natural Logarithm

Take the natural logarithm on both sides to utilize properties that allow us to work with exponents. This gives us: \( \ln(e^{5t}) = \ln(k^t) \).

Step 3: Use Logarithm Properties

Using the property of logarithms that \( \ln(a^b) = b\ln(a) \), we have:\( 5t = t \ln(k) \).

Step 4: Solve for \( \ln(k) \)

Divide both sides of the equation by \( t \) (assuming \( t eq 0 \)) to isolate \( \ln(k) \):\( \ln(k) = 5 \).

Step 5: Exponentiate to Solve for \( k \)

Exponentiate both sides to solve for \( k \):\[ k = e^5 \].

Key Concepts

Natural Logarithm

The natural logarithm is an important mathematical concept used to solve equations involving exponentials. When you see \( \ln \), it refers to the natural logarithm, which is the logarithm to the base of Euler's number \( e \). Euler's number \( e \) is an irrational number approximately equal to 2.71828.

• The natural logarithm \( \ln(x) \) is used specifically when working with exponentials involving \( e \).
• It is the inverse operation of exponentiation with base \( e \). For example, if \( e^a = x \), then \( \ln(x) = a \).

Applying the natural logarithm is very useful because it transforms a multiplicative relationship into an additive one, which is typically easier to handle. In the original problem, when we take the natural logarithm on both sides of the equation \( e^{5t} = k^t \), it has the effect of bringing the exponent down and creating a more manageable equation: \( 5t = t \ln(k) \). This transformation is vital for solving equations involving exponents.

Logarithm Properties

Logarithms have several key properties that are extremely helpful when solving equations where unknowns are found as exponents. One of the most important properties is the power rule:

• Power Rule: \( \ln(a^b) = b\ln(a) \) allows you to move the exponent to the front of the logarithm.
• This property transforms complex exponential expressions into more straightforward linear equations.

In the given exercise, we use this property with \( \ln(e^{5t}) \) where the exponent \( 5t \) is brought in front, resulting in \( 5t \ln(e) \). Since \( \ln(e) = 1 \), this simplifies further to \( 5t \).
Similarly, applying the property to \( \ln(k^t) \) gives \( t \ln(k) \). By utilizing these properties, exponential equations can be easily manipulated to solve for the unknown variable. This eliminates the complexity of solving non-linear equations.

Exponentiation

Exponentiation involves expressions where a number, called the base, is raised to the power of an exponent. In equations, exponentiation can often create equations that are difficult to solve using standard algebraic techniques. Thus, understanding how to undo exponentiation is very useful.

• In the context of logarithms: Taking the logarithm of an exponent allows you to "bring down" the exponent for easier manipulation.

After simplifying an equation like \( 5t = t \ln(k) \) using logarithm properties, you can solve for \( \ln(k) \). Once \( \ln(k) = 5 \) is established, we "undo" the logarithm by exponentiating both sides to solve for \( k \). This involves raising Euler's number \( e \) to both sides:
\[k = e^5\]
This step is crucial because it provides the solution to the original problem, confirming \( k \) as \( e^5 \). Exponentiation is therefore not only a method for expanding an expression but is also a crucial step in solving exponential equations through the use of logarithms.