Problem 55 - 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. $$ y=2.5(0.7)^{x} $$

Step-by-Step Solution

Answer

The equation \(y=2.5(0.7)^{x}\) can be rewritten in terms of base \(e\) and expressed in terms of a natural logarithm as \(x = \frac{\ln (y/2.5)}{\ln 0.7}\). If a particular \(y\) value is provided, substitute it into the equation to calculate the corresponding \(x\) value, simplified and rounded to three decimal places.

Step 1: Write the given equation in logarithmic form

To begin, wishing to rewrite the given equation, \(y=2.5(0.7)^{x}\) in logarithmic form, it can be written as \(\log_{0.7}(y/2.5) = x\) where the base is 0.7.

Step 2: Use the change of base formula to switch to base \(e\)

Logarithm change of base formula, which states that \(\log_{b}a = \frac{\ln a}{\ln b}\) can be applied to switch the base to \(e\). Hence, equation \(\log_{0.7}(y/2.5) = x\) can be re-written as \(x = \frac{\ln (y/2.5)}{\ln 0.7}\). This represents \(x\) in terms of a natural logarithm where base of natural logarithm is \(e\).

Step 3: Simplify and round to three decimal places

If a specific value of \(y\) is provided, the numerical value of \(x\) can be found by substituting \(y\) in the equation obtained from Step 2, which can then be further simplified and rounded to the nearest three decimal places.

Key Concepts

Change of Base Formula

When dealing with logarithms, you might encounter different bases, such as base 10 (common logarithm) or base \(e\) (natural logarithm). If you want to convert between these two, you can use the change of base formula. This formula is a handy tool that allows you to switch from one logarithmic base to another with ease.

The change of base formula is given by:

• \(\log_{b}a = \frac{\ln a}{\ln b}\)

Here, \(\ln\) represents the natural logarithm, which uses base \(e\). This formula helps in solving logarithmic equations where the base is not readily available on standard calculators.
For example, in our exercise, we need to rewrite the expression \(\log_{0.7}(\frac{y}{2.5}) = x\) using base \(e\). By applying the change of base formula, we can express it as:

• \(x = \frac{\ln (y/2.5)}{\ln 0.7}\)

This transforms the equation into a form that is compatible with natural logarithms and simplifies further manipulation.

Rewriting Equations

Rewriting an equation involves transforming it into a preferred form for easier analysis or computation. In mathematics, precise transformations often involve changing the form but not the fundamental meaning of an equation.

For instance, in the given exercise, we start with the exponential equation:

• \(y=2.5(0.7)^{x}\)

This type of equation can be rewritten in logarithmic form to highlight the relationship of \(x\) in terms of \(y\) and a given base, thereby making it easier to handle when solving for specific values of \(x\). The equation becomes:

• \(\log_{0.7}(y/2.5) = x\)

By transforming the exponential equation into its logarithmic counterpart, you can apply other mathematical techniques such as the change of base formula, simplifying the solution process.

Rounding Decimals

Rounding is a method to simplify numbers to make them easier to work with or understand. For instance, rounding to three decimal places means adjusting the number to the nearest thousandth.

Once we compute \(x\) using the natural logarithm equation \(x = \frac{\ln (y/2.5)}{\ln 0.7}\), the result is often a number with more than three decimals. To make it simpler, and if a situation demands, you should round it.
For example:

• If \(x = 1.234567\), rounding to three decimal places gives \(x = 1.235\)

This practice is essential in real-world applications, such as finances, where exact precision isn't always necessary, but ease of interpretation is. Rounding helps make the numbers manageable and communicates the precision level used in calculations.