Problem 32 - College Algebra

Question

Rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm, and then round to three decimal places. $$ y=1000(7.3)^{x} $$

Step-by-Step Solution

Answer

The equation rewritten in terms of base \(e\) and expressed in terms of natural logarithm is \(y=1000e^{ln(7.3^x)}\) or \(y=1000*(7.3^x)\)

Step 1: Write the original expression

Start with the equation provided in the exercise: \(y=1000(7.3)^{x}\).

Step 2: Convert to natural base

To rewrite this in terms of base \(e\) using natural logarithm, use the base change formula, which tells us that \(log_b(a)=ln(a)/ln(b)\). By substituting \(b=7.3\) and \(a=7.3^x\), we can see that \(log_{7.3}(7.3^x)=x=log_e(7.3^x)/log_e(7.3)\). Multiplying both sides by \(log_e(7.3)\), we get \(xlog_e(7.3)=log_e(7.3^x)\). So, the given equation can be rewritten: \(y=1000e^{xln(7.3)}\).

Step 3: Simplify the expression

From step 2, we have the equation in terms of base \(e\). Simplify the expression in the exponent to give the final simplified expression: \(y=1000e^{ln(7.3^x)}\). Given that \(e^{ln(x)} = x\), we can simplify further to give: \(y=1000*(7.3^x)\).

Key Concepts

Base Change Formula

The base change formula is an invaluable tool when dealing with logarithms, particularly when you need to switch from one logarithmic base to another. The formula is given by:

• \(\log_b(a) = \frac{\ln(a)}{\ln(b)}\)

This allows you to express a logarithm in any base \(b\) in terms of natural logarithms (\(\ln\)). The reason why this conversion is useful is because natural logarithms and base \(e\) are commonly used in mathematics, especially in calculus and exponential growth problems.
To see how the base change formula works in practice, consider the need to express \(7.3^x\) in terms of base \(e\). This is accomplished by expressing the logarithm as \(\ln(7.3^x)/\ln(7.3)\), which simplifies to \(x\ln(7.3)\) when rearranged. This conversion then permits the exponential expression to be re-written conveniently with \(e\) as the base, facilitating further computation or simplification.

Exponential Functions

Exponential functions are equations where the variable appears in the exponent. A general form for exponential functions is \(y = a(b^x)\), where \(a\) is a constant, \(b\) is the base of the exponential function, and \(x\) is the exponent.

• They are key in modeling growth or decay processes, like population growth or radioactive decay.
• The base \(b\) is particularly important, as it determines how fast the growth or decay occurs.

When considering converting exponential functions to base \(e\), it's useful to leverage the properties of natural logarithms, as seen in our exercise. Here, an expression such as \(7.3^x\) was rewritten as \(e^{x\ln(7.3)}\) by applying the base change formula. By using a natural base, complex exponential expressions become more manageable, allowing easier interpretation and manipulation, especially in calculus.

Natural Base e

The natural base \(e\) is a mathematical constant approximately equal to 2.71828. It is fundamental in the field of mathematics due to its unique properties, particularly in calculus and exponential models.

• The number \(e\) arises naturally from the process of mathematical growth and decay.
• Expressions with \(e\) are easier to manipulate in calculus because of their natural relationship with differentiation and integration.

In our context, rewriting functions in terms of base \(e\) simplifies the handling of exponential equations. The expression \(1000(7.3)^x\) was converted into \(1000e^{x\ln(7.3)}\). This rewriting aligns the function with the properties of \(e\), making derivative calculations and integration more intuitive. As such, converting to base \(e\) not only simplifies mathematical manipulation but also enhances comprehension of the underlying exponential relationships.