Problem 54 - Algebra and Trigonometry

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, when rewritten in terms of base e, and expressed as a natural logarithm is \( x = ln(y/1000) / ln(7.3) \). Due to the nature of the question, it's not possible to provide a numerical answer rounded to three decimal places as y is not specified.

Step 1: Rewriting Equation to have Exponential Isolated

First, let's adjust the equation \( y = 1000(7.3)^{x} \) to isolate the exponent by dividing both sides by 1000. This simplifies the equation to \(y/1000 = (7.3)^{x}\). This is an important step because it puts the equation in a more convenient form for conversion to logarithmic form.

Step 2: Converting to Natural Logarithmic Form

Next, we convert the simplified equation from exponential form to natural logarithmic form. An exponential equation \( b^{x} = a \) can be converted to a logarithmic equation \( log_{b}(a) = x \). Hence, the equation becomes \( ln(y/1000) = x*ln(7.3) \).

Step 3: Solving for x

Finally, we solve the logarithmic equation for x. This is done by dividing both sides of the equation by \( ln(7.3) \), hence: \( x = ln(y/1000) / ln(7.3) \).

Key Concepts

Exponential Form

The exponential form is a way of expressing numbers that shows how many times a base number is multiplied by itself. In the equation from the exercise, the format is important because it contains a base and an exponent. Here, the equation is given as \( y = 1000(7.3)^x \). The term \( (7.3)^x \) is in exponential form, where 7.3 is the base raised to the power of \( x \) — the exponent. This structure allows for the representation of large numbers or complex calculations in a concise manner. To simplify solving processes, especially conversions, it often requires isolating this exponential part by handling the equation algebraically, such as dividing by constants like 1000 in the given example.

Logarithmic Equation

A logarithmic equation is a mathematical expression that relates logarithms of numbers or expressions. It serves as the inverse operation to exponentiation, which means it helps in solving equations involving exponents by "undoing" the exponential process. To convert an exponential equation into a logarithmic one, recognize that \( b^x = a \) transforms into \( \log_b(a) = x \).

• This implies that if you have a value, raised to the unknown power, equal to another value, the logarithmic equation will express the exponent (unknown power) as a function of the two known values.
• In the step-by-step solution provided, converting \((7.3)^x = y/1000\) gives us \( \ln(y/1000) = x \ln(7.3) \).

Logarithmic equations such as these enable us to solve for the unknown exponent, making them extremely useful in modeling exponential decay or growth.

Base e

The base \( e \) is a fundamental mathematical constant approximately equal to 2.71828. It is the base of the "natural logarithm," denoted as \( \ln \).

• The beauty of using \( e \) as the base is seen in calculus, where the functions involving \( e \) have unique properties such as having a rate of growth proportional to their current size.
• In the context of the problem, rewriting the equation in terms of base \( e \) allows the equation to be represented with natural logarithms, which possess simpler and well-understood algebraic properties.

This conversion is crucial in many fields, including finance, where \( e \) is used to calculate continuously compounded interest, and in various scientific applications.

Conversion to Logarithmic

Conversion to logarithmic form involves changing an expression from exponential notation to a logarithmic notation. This is an essential skill because it allows for easier manipulation and solution of equations involving exponentials.

• From the exercise, the isolated exponential form \( y/1000 = (7.3)^x \) was converted to logarithmic form \( \ln(y/1000) = x \ln(7.3) \).
• This process involves applying logarithms to both sides of the equation to solve for the exponent \( x \).

This conversion simplifies complex expressions and makes solving equations possible using logarithm properties, for instance, \( \log_b(mn) = \log_b(m) + \log_b(n) \). Making the conversion back and forth between these forms is crucial for understanding and analyzing exponential models correctly.