Problem 53 - College Algebra

Question

In Exercises \(53-56,\) rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm and then round to three decimal places. $$y=100(4.6)^{x}$$

Step-by-Step Solution

Answer

The equation rewritten in terms of base \(e\) and expressed as a natural logarithm is: \(x = \frac{\ln y - \ln 100}{\ln 4.6}\).

Step 1: Convert the base of the exponent

According to the formula, rewrite the expression \(4.6^x\) as \(e^{x \ln 4.6}\). So, the equation becomes: \(y = 100e^{x \ln 4.6}\).

Step 2: Write equation in natural logarithm form

Take natural logarithm on both sides of the equation to eliminate the base \(e\) from the equation: \(\ln y = \ln (100e^{x \ln 4.6})\). Then, use the property of logarithms \(\ln ab = \ln a + \ln b\), the equation becomes: \(\ln y = \ln 100 + \ln e^{x \ln 4.6}\). Then, use the identity \(\ln e^x = x\) for the second term, resulting in: \(\ln y = \ln 100 + x \ln 4.6\).

Step 3: Solve for \(x\)

To solve for \(x\), subtract \(\ln 100\) from both sides of the equation: \(x \ln 4.6 = \ln y - \ln 100\). Finally, divide both sides by \(\ln 4.6\) to solve for \(x: x = \frac{\ln y - \ln 100}{\ln 4.6}\).

Key Concepts

Exponential Equations

Exponential equations involve expressions where variables occur as exponents. They can look intimidating, but they're not so complicated once you get the hang of them.
In an exponential equation, the general form is \( a^{x} = b \), where \( a \) is the base, \( x \) is the exponent, and \( b \) is the resulting value. Solving these equations typically requires you to match the bases on both sides.

• If the equation is already similar to the form \( a^{x} = a^{y} \), then you can simply set \( x = y \).
• If the bases are different, you might need to use logarithms to solve for the exponent.

For instance, in the expression \( 100(4.6)^x \), the challenge is converting it to involve the natural exponential base \( e \), which simplifies the use of logarithms.

Logarithmic Properties

Logarithms are incredibly useful tools in mathematics for manipulating exponents. They can simplify complex equations by turning multiplication into addition and exponents into coefficients. These properties make solving exponential equations much simpler. Here are some key logarithmic properties:

• Product Rule: \( \ln(ab) = \ln a + \ln b \)
• Quotient Rule: \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \)
• Power Rule: \( \ln(a^x) = x\ln a \)
• Identity: \( \ln (e) = 1 \)

Understanding these properties allows you to transform an equation like \( \ln (100e^{x \ln 4.6}) \) into a simpler form, \( \ln 100 + x \ln 4.6 \). Knowing how to apply these rules effectively provides a solid foundation for solving logarithmic and exponential equations.

Base e Conversion

Base \( e \) conversion relates to expressing numbers in terms of natural logarithms, which is especially helpful with exponential functions. The number \( e \) (approximately equal to 2.718) is a fundamental constant in mathematics, known as Euler's number. Using \( e \) simplifies calculus operations and provides a natural progression in growth models.
The key to converting to base \( e \) involves understanding that any number \( a \) can be rewritten in the form \( e^{\ln a} \). For example, in converting the base of \( 4.6 \) to \( e \), we write it as \( e^{\ln 4.6} \). Consequently, the expression \( 4.6^x \) converts to \( e^{x \ln 4.6} \).
This conversion is crucial because it allows you to leverage the properties of natural logarithms (\( \ln \)) when solving equations. Thus, once a problem is rewritten in terms of base \( e \), we can easily apply logarithmic properties for a straightforward solution.