Problem 7 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

\(t = \ln(10) \approx 2.302\)

Step 1: Understand the Equation

The equation given is \(10 = e^t\), which is an exponential equation. Our task is to solve for the value of \(t\).

Step 2: Take the Natural Logarithm of Both Sides

To isolate \(t\), we take the natural logarithm (\(\ln\)) on both sides of the equation to eliminate the exponent. This gives us \(\ln(10) = \ln(e^t)\).

Step 3: Apply the Logarithmic Identity

Use the logarithmic identity \(\ln(a^b) = b\ln(a)\). For \(\ln(e^t)\), apply this identity to get \(t\ln(e)\). Since \(\ln(e) = 1\), this simplifies to \(t\).

Step 4: Solve for t

After applying the logarithmic identity, the equation becomes \(\ln(10) = t\). This means \(t = \ln(10)\).

Step 5: Calculate the Value of t

Finally, compute the natural logarithm of 10 using a calculator to find the numerical value: \(t \approx 2.302\).

Key Concepts

Exponential Functions

Exponential functions are central to various scientific and mathematical fields. They involve equations where a variable appears in the exponent, like \( y = e^x \). The base of the function, \( e \), is approximately 2.718, known as Euler's number. This number is crucial in natural growth processes and complex calculations related to growth and decay.

These functions work based on the principle that with each unit increase in the exponent, the value of the function multiplies by the base. This creates a situation where exponential functions rapidly increase or decrease.

• For example, in the equation \( 10 = e^t \), \( t \) is the exponent that we need to determine.
• Exponential equations often appear in real-world scenarios like population growth and radioactive decay.

Exponential functions are versatile and appear in various fields such as biology, physics, and finance.

Logarithmic Identities

Logarithmic identities are mathematical rules that help simplify expressions involving logarithms. These rules make it easier to solve equations where variables are exponents.

The natural logarithm, denoted as \( \ln \), is the inverse of the exponential function \( e^x \). Logarithmic identities include rules like \( \ln(a^b) = b\ln(a) \), which allow us to move the exponent in a logarithmic expression. This identity is vital in solving equations like \( \ln(e^t) \).

• Using \( \ln(e^t) = t\ln(e) \) and knowing \( \ln(e) = 1 \), we simplify expressions easily.
• These identities are particularly useful because they transform multiplication into addition, division into subtraction, and power into multiplication, aiding problem-solving.

Understanding these identities is key to handling complex logarithmic and exponential functions.

Solving Exponential Equations

Solving exponential equations using natural logarithms can be straightforward once you know the process. It typically involves isolating the exponential term and applying logarithms to simplify. Let's see how this works step by step:

1. **Isolate the Exponential Term**: Consider the equation \( 10 = e^t \). Here, we want \( t \) by itself, so we need to get rid of \( e \).

• We achieve this by using the natural logarithm \( \ln \), as it is the inverse operation of exponentiation with base \( e \).

2. **Apply the Natural Logarithm**: Take the \( \ln \) of both sides, giving you \( \ln(10) = \ln(e^t) \).

• Use the logarithmic identity mentioned earlier to simplify this to \( \ln(10) = t \).
• After simplifying, you find \( t = \ln(10) \).

3. **Calculate the Result**: Plug \( 10 \) into your calculator's natural logarithm function to find \( t \). For example, \( t \approx 2.302 \).

When you need to solve exponential equations quickly and accurately, understanding and using the right processes and identities makes things much easier.