Problem 12 - Applied Calculus

Question

For Problems \(1-16,\) solve for \(t\) using natural logarithms. $$a=b^{t}$$

Step-by-Step Solution

Answer

\( t = \frac{\ln(a)}{\ln(b)} \)

Step 1: Understand the Equation

We are given the equation \( a = b^t \). This means \( a \) equals \( b \) raised to the power of \( t \). We need to solve this equation for \( t \).

Step 2: Take the Natural Logarithm on Both Sides

To solve for \( t \), we need to eliminate the exponent. We do this by taking the natural logarithm (ln) of both sides of the equation. Thus, we have: \( \ln(a) = \ln(b^t) \).

Step 3: Apply the Logarithmic Power Rule

The logarithmic power rule states that \( \ln(b^t) = t \cdot \ln(b) \). Apply this rule to the right side: \[ \ln(a) = t \cdot \ln(b) \].

Step 4: Solve for t

To isolate \( t \), divide both sides of the equation by \( \ln(b) \), resulting in: \[ t = \frac{\ln(a)}{\ln(b)} \].

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. In the equation \(a = b^t\), the base is \(b\) and the exponent is \(t\). Exponentiation is fundamentally about repetition.
The base, \(b\), is multiplied by itself \(t\) times.
A few key characteristics of exponentiation include:

• It is not commutative, meaning \(b^t\) is not the same as \(t^b\).
• It is associative with respect to multiplication, such that \((b^t)^s = b^{(t\cdot s)}\).
• The exponent can be any real number, which allows for operations involving roots and fractional exponents.

In our problem, \(a\) represents the outcome of raising \(b\) to \(t\). Understanding this relationship is key when solving equations where the variable is in the exponent.

Logarithmic Power Rule

The logarithmic power rule is a useful tool when dealing with equations where the variable appears as an exponent. The rule provides a way to "bring down" the exponent, making the equation easier to solve.
To apply this rule, you start with the logarithm of an exponentiated number, for instance, \(\ln(b^t)\). The rule states:

• The logarithm of a power, \(b^t\), is the exponent times the logarithm of the base.

This is mathematically expressed as \(\ln(b^t) = t \cdot \ln(b)\).
This transformation is critical as it reduces the problem from an exponential equation to a linear equation, which is far simpler to manipulate. Recognizing and properly applying the logarithmic power rule is a fundamental skill in solving many logarithmic equations.

Solving Logarithmic Equations

Solving logarithmic equations often involves multiple steps and a comprehensive understanding of how logarithms and exponents interact. When you have an equation like \(a = b^t\), you can solve it by following these steps:

• Take the natural logarithm: Start by taking the \(\ln\) of both sides, transforming it into \(\ln(a) = \ln(b^t)\).
• Apply the logarithmic power rule: Use this rule to rewrite \(\ln(b^t)\) as \(t \cdot \ln(b)\).
• Isolate the variable: Solve for \(t\) by dividing both sides by \(\ln(b)\).

This process results in the solution \(t = \frac{\ln(a)}{\ln(b)}\).
Logarithmic equations often appear complex due to their format, but applying the rules consistently will lead you to a solution. Always double-check your steps, as each transformation must be correct to maintain the equation's equality.