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

Step-by-Step Solution

Answer

The rewritten form of the function in terms of the base \(e\) and rounded to three decimal places is \[\ln y = 0.916 - 0.357x\].

Step 1: Original equation

The given equation is \[y = 2.5 \cdot 0.7^{x}\] The goal is to rewrite this equation in terms of the base \(e\) to express the answer in terms of a natural logarithm.

Step 2: Use logarithmic properties

Recall the change of base formula for logarithms, which states that\[\log_b a = \frac{\log_c a}{\log_c b}\],where \(c\) could be any base, but in this case, we want to convert to base \(e\), so we will use the natural logarithm (\(\ln\)). Taking the natural logarithm of both sides of the equation results in\[\ln y = \ln(2.5 \cdot 0.7^{x})\],which can be simplified (using the logarithm property \(\ln ab = \ln a + \ln b\)) to\[\ln y = \ln 2.5 + \ln 0.7^{x}\].

Step 3: Apply power property of logarithms

Use the power rule for logarithms \[\ln a^n = n \ln a\]to move the exponent \(x\) in front of the logarithm:\[\ln y =\ln 2.5 + x \cdot \ln 0.7\]. This gives us the equation in terms of \(\ln\), or base \(e\).

Step 4: Round to three decimal places

For the final step, round \(\ln 2.5\) and \(\ln 0.7\) to three decimal points. This leads to \[\ln y = 0.916 + x \cdot (-0.357)=0.916 - 0.357x\].

Key Concepts

Exponential Functions

Exponential functions are a powerful mathematical tool used to model situations where a quantity grows or decays at a constant rate. These functions are written in the form \( y = a \cdot b^x \), where \( a \) represents the initial amount, \( b \) is the base showing the growth (if greater than 1) or decay (if less than 1) factor, and \( x \) is the exponent. In our problem, we dealt with the function \( y = 2.5 \cdot 0.7^x \). Here, the coefficient \( 2.5 \) is the initial quantity, and \( 0.7 \) is the factor by which \( y \) decreases with each unit increase of \( x \). The smaller the base, the more quickly \( y \) decreases. Exponential functions frequently pop up in sciences to describe phenomena like population growth, radioactive decay, and more.

Logarithmic Properties

Logarithmic properties make it easier to manipulate complex expressions, especially when involving exponential equations. These properties stem from the definition that a logarithm is the inverse function of exponentiation. Essential properties include:

• Product Property: \( \log_b(xy) = \log_b x + \log_b y \)
• Quotient Property: \( \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \)
• Power Property: \( \log_b(x^y) = y \cdot \log_b x \)

These properties help simplify expressions and solve for unknowns in equations. By breaking down the original equation into simpler parts, solving becomes more straightforward. For instance, by utilizing the product property, the original exercise's expression \( \ln(2.5 \cdot 0.7^x) \) became \( \ln 2.5 + \ln 0.7^x \).

Change of Base Formula

The change of base formula can convert logarithms from one base to another. This is particularly useful when certain bases (like base \( e \) for natural logarithms) are more easily handled in calculators or contexts involving compound growth rates. The formula reads \( \log_b a = \frac{\log_c a}{\log_c b} \), where \( c \) is the new base. In many situations, we convert to base \( e \) due to its natural properties and prevalence in calculus. For the given exercise, we used this concept. We turned expressions involving \( 0.7 \) and \( 2.5 \) into natural logs since this format often simplifies solving real-world problems, where exponential decay rates are modeled with \( e \).

Power Rule for Logarithms

The power rule for logarithms simplifies expressions with exponents inside the logarithmic function. This rule states that \( \log_b(x^n) = n \cdot \log_b x \). The exponent \( n \) moves to the front, making the expression much easier to evaluate. This rule was utilized in the step-by-step solution of our problem, where \( \ln 0.7^x \) was simplified using the power rule to become \( x \cdot \ln 0.7 \). This transformation allows us to express complicated exponential equations linearly in terms of \( x \), simplifying calculations and highlighting relationships between variables. This powerful technique makes logs an essential tool in many mathematical applications.