Problem 56 - 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. $$ y=4.5(0.6)^{x} $$

Step-by-Step Solution

Answer

The equation written in a natural logarithm of base e, rounded to three decimal places is \( x= \frac{ln ( \frac{y}{4.5})}{ln 0.6} \). Read as, 'x equals the natural logarithm of y divided by 4.5, all divided by the natural logarithm of 0.6'. The final numerical result depends on the given value of y.

Step 1: Write the equation in logarithmic form

The expression, \(y = 4.5(0.6)^x\), can be written as a logarithmic equation by applying the logarithmic identity that the logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number. So, we can write it as \( \log_{0.6}{ \frac{y}{4.5}} = x \)

Step 2: Change the base to 'e'

Now, we employ the change of base formula, which states that \( \log_a b = \frac{\log_c b}{\log_c a} \), where c is the new base and in this case, c=e (approximately equals 2.71828). Using this formula, our equation becomes \( x= \frac{\log_e ( \frac{y}{4.5})}{\log_e 0.6} \). The \(\log_e\) function is more commonly represented as the natural logarithm \( ln \), so we rewrite it as \( x= \frac{ln ( \frac{y}{4.5})}{ln 0.6} \)

Step 3: Round the answer

Round off the result obtained in Step 2 to the nearest three decimal places. Therefore, the final answer will be expressed in terms of y. However, without a specified value for y, we cannot provide a rounded numerical result at this step. For practical purposes, the value of y can be substituted into this equation to find the corresponding x value, rounded to three decimal places.

Key Concepts

Change of Base Formula

The change of base formula is a helpful tool in mathematics when you want to switch between different logarithmic bases. This is especially useful when you're dealing with uncommon bases. The formula is:

• \( \log_a b = \frac{\log_c b}{\log_c a} \)

Here, \(a\) is the old base that you're changing from, \(b\) is the number you're taking the logarithm of, and \(c\) is the new base you're switching to. Often, in mathematical problems, we prefer using the natural base \(e\), which is approximately 2.71828, leading to the natural logarithm notated as \(\ln\). So, using this formula, you can convert any logarithm to base \(e\) by setting \(c = e\). This is exactly what we did in the given exercise, changing the base of the logarithmic expression from 0.6 to \(e\). This ability to change base makes calculations more manageable, especially when dealing with calculators or logarithm tables that typically provide only common logarithms (base 10) or natural logarithms (base \(e\)).

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, which we call the base, to a power, termed as the exponent.

• For example, in the expression \((0.6)^x\), 0.6 is the base, and \(x\) is the exponent.

This means that the base, 0.6, is multiplied by itself \(x\) times. Exponentiation is the inverse operation of logarithms. While exponentiation is crucial for modeling growth and decay processes, it can sometimes be challenging to work with. That's where logarithms come in handy! By transforming an exponential equation like \(y = 4.5(0.6)^x\) into its logarithmic form, it simplifies solving for unknowns like \(x\). Understanding exponentiation is foundational not only for this exercise but also for various applications in science and engineering.

Logarithmic Equation

A logarithmic equation is an equation that involves logarithms, often used to solve for a variable that appears as an exponent. In simpler terms, it's when you're looking for the power to which a base number has been raised. The equation \(\log_{0.6}{\frac{y}{4.5}} = x\) is a prime example. This was derived from the original exponential equation \(y = 4.5(0.6)^x\) by applying logarithmic transformation. One might wonder why we wanted a logarithmic form. Well, logarithmic equations allow us to transfer the exponential from the right-hand side to the left using the power property of logarithms, converting a multiplication operation into an addition operation. This is particularly convenient for solving equations where the unknown variable is the exponent, making complex calculations simpler and more approachable.

Rounding Numbers

Rounding numbers is an essential skill used in mathematics to make figures easier to work with and to present results in a simplified manner. Often, answers can be precise to many decimal places, and it's not practical or necessary to use all those digits.

• In our case, rounding to three decimal places means you'll look at the fourth digit after the decimal point. If it's 5 or larger, you'll round up, otherwise, you'll round down.

For example, if the number is 2.7184, it rounds to 2.718 because the fourth digit (4) is less than 5. On the other hand, if we had 2.7185, we'd round it to 2.719. Rounding makes handling data less cumbersome and often more comprehensible, especially when interpreting results in scientific and business contexts. It also allows standardization across calculations, as seen in the exercise where the final form is requested to be rounded to three decimal places.