Problem 4 - Applied Calculus

Question

Solve for \(t\) using natural logarithms. $$130=10^{t}$$

Step-by-Step Solution

Answer

The value of \( t \) is approximately 2.1146.

Step 1: Understand the Equation

The equation given is in the form of an exponential function: \( 130 = 10^t \). Here, we need to solve for the exponent \( t \).

Step 2: Apply the Natural Logarithm

To isolate \( t \), we apply the natural logarithm (\( \ln \)) to both sides of the equation: \( \ln(130) = \ln(10^t) \).

Step 3: Use the Property of Logarithms

Use the property of logarithms, \( \ln(a^b) = b \ln(a) \), to bring the exponent \( t \) down: \( \ln(130) = t \ln(10) \).

Step 4: Isolate \( t \)

Divide both sides of the equation by \( \ln(10) \) to solve for \( t \): \( t = \frac{\ln(130)}{\ln(10)} \).

Step 5: Calculate the Value of \( t \)

Use a calculator to compute \( \ln(130) \) and \( \ln(10) \), and perform the division: \( t \approx \frac{4.8675}{2.3026} \approx 2.1146 \).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. These functions have the form \( y = a^x \), where \( a \) is a positive constant known as the base and \( x \) is the exponent. Exponential functions are widely used in various fields such as biology, finance, and physics because they can model growth or decay processes efficiently.
Key characteristics of exponential functions include:

• The base \( a \) is always a positive real number. When \( a = 1 \), the function is constant.
• If \( a > 1 \), the function represents exponential growth. As \( x \) increases, \( y \) rises rapidly.
• If \( 0 < a < 1 \), the function represents exponential decay. As \( x \) increases, \( y \) decreases towards zero.
• The graph of an exponential function is smooth and continuous, and it will never touch the x-axis, making the x-axis a horizontal asymptote.

Understanding exponential functions helps in preparing us for their inverse operations, logarithms, which are vital in solving exponential equations.

Logarithmic Properties

The concept of logarithms is closely tied to exponentials. A logarithm answers the question: "To what exponent must the base be raised, to produce a given number?" For example, in the expression \( \log_b(x) = y \), \( b^y = x \).
There are a few crucial properties of logarithms that help in manipulating and solving exponential equations:

• Product Property: \( \log_b(MN) = \log_b(M) + \log_b(N) \)
• Quotient Property: \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \)
• Power Property: \( \log_b(M^p) = p\log_b(M) \)
• Change of Base Formula: \( \log_b(M) = \frac{\log_k(M)}{\log_k(b)} \) for any positive base \( k \)

These properties allow us to simplify complex logarithmic expressions and are instrumental in solving equations involving exponents. Specifically, the Power Property is used to "bring down" exponents to solve for variables.

Solving Exponential Equations

Solving exponential equations often involves taking logarithms. When faced with an equation like \( a^x = b \), the goal is to isolate the variable \( x \). This is where natural logarithms come in handy.
Natural logarithms use the base Euler’s number \( e \) (approximately 2.718), and denoted as \( \ln \). Although we could use any logarithm base to solve an exponential equation, natural logarithms are particularly useful due to their frequent application in calculus and natural growth processes.
To solve an exponential equation using natural logarithms, follow these straightforward steps:

• First, take the natural logarithm of both sides of the equation. This transforms the equation: \( a^x = b \) becomes \( \ln(a^x) = \ln(b) \).
• Next, apply the Power Property of logarithms to move the exponent down: \( x\ln(a) = \ln(b) \).
• Finally, solve for \( x \) by dividing both sides by \( \ln(a) \): \( x = \frac{\ln(b)}{\ln(a)} \).

This method not only simplifies the process but highlights the connection between exponentials and logarithms, offering a reliable toolset for handling various mathematical challenges.