Problem 12 - Applied Calculus

Question

Solve for \(t\) using natural logarithms. $$a=b^{t}$$

Step-by-Step Solution

Answer

Solution: \( t = \frac{\ln(a)}{\ln(b)} \) assuming \(a > 0\) and \(b > 0, b \neq 1\).

Step 1: Understand the Problem

The task is to solve the equation \(a = b^t\) for the variable \(t\). This involves isolating \(t\) in the equation.

Step 2: Take Natural Logarithm of Both Sides

Apply the natural logarithm (ln) to both sides of the equation to make use of the logarithmic identity that helps in dealing with exponents. This gives us: \[ \ln(a) = \ln(b^t) \]

Step 3: Use the Power Rule for Logarithms

Utilize the logarithmic power rule which states \(\ln(b^t) = t \cdot \ln(b)\). Substitute this back into the equation:\[ \ln(a) = t \cdot \ln(b) \]

Step 4: Solve for t

Isolate \(t\) by dividing both sides of the equation by \(\ln(b)\), assuming \(b > 0\) and \(b eq 1\):\[ t = \frac{\ln(a)}{\ln(b)} \]

Step 5: Verify Domain Constraints

Ensure that the solution is valid by checking the domain constraints. Both \(a\) and \(b\) must be positive real numbers because the natural logarithm is defined only for positive values.

Key Concepts

Natural Logarithms

Natural logarithms are a specific type of logarithm that use the base "e". The number "e" is an irrational constant approximately equal to 2.71828. Natural logarithms are denoted as "ln". When solving equations that involve exponential terms, natural logarithms can often simplify the process because they directly relate to the base of natural growth (e). Whenever you see an equation of the form like \(a = b^t\), applying a natural logarithm can transform a complicated exponent into something much more manageable. This transformation is due to the properties of logarithms, which allow us to bring the exponent down to the same level as the other numerical factors in the equation. Using natural logarithms in equations is particularly handy because

• They convert multiplication into addition.
• They help in solving exponentials.
• They are suitable for equations involving continuous growth and decay.

Logarithmic Identities

Logarithmic identities are essential tools in mathematics that allow us to manipulate and simplify logarithmic expressions. One crucial logarithmic identity is the one used to bring down an exponent. These are two basic identities that are incredibly useful:

• \(\log(b^t) = t \cdot \log(b)\) – This allows the exponent "t" to be pulled down as a multiplier.
• \(\log(a \cdot b) = \log(a) + \log(b)\) – This breaks down a product into a sum.

In our problem, we used the first identity by taking the natural logarithm of both sides of the equation \(a = b^t\). This action allowed us to exploit the identity by converting the power into a product, simplifying the equation into \(\ln(a) = t \cdot \ln(b)\). This greatly simplifies isolating and solving for the exponent "

Power Rule for Logarithms

The Power Rule for Logarithms states that the logarithm of a power is equal to the exponent times the logarithm of the base. Mathematically, this is expressed as \(\ln(b^t) = t \cdot \ln(b)\). The Power Rule is one of the core tools when dealing with exponential equations. In our problem, the equation \(a = b^t\) was transformed by taking the natural logarithm on both sides, This rule simplifies the solving process by:

• Turning exponential growth or decay into linear relationships.
• Makes solving for the variable exponent much more straightforward.

By applying the Power Rule, we were able to write the equation as a linear relationship between \(t\) and the other terms, thereby making it easier to solve for \(t\).

Domain Constraints

Domain constraints refer to the values that variables can assume in any given mathematical expression or equation. In the context of logarithmic equations, specific constraints ensure that the expressions remain valid. When using natural logarithms, it's critical to remember that:

• Both \(a\) and \(b\) must be positive real numbers.
• The base \(b\) should not be equal to 1, as it undermines the purpose of finding a meaningful exponent.

These constraints are essential because the logarithm is undefined for non-positive values, which would result in invalid and nonsensical equations. Ensuring that \(a\) and \(b\) adhere to these conditions guarantees that our derived solution \(t = \frac{\ln(a)}{\ln(b)}\) is meaningful and mathematically valid. By carefully considering the domain constraints, we confirm that our operations and final result hold true within the permissible boundaries of the problem at hand.