Problem 34 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each exponential equation by taking the logarithm on both sides. Express the solution set in terms of logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$135-(4.7)^{x}=0$$

Step-by-Step Solution

Answer

The exact solution to the equation in terms of logarithms is \(x = \frac{ln(135)}{ln(4.7)}\), and the decimal approximation to two decimal places is \(x ≈ 3.76\).

Step 1: Rearrange the equation

First, the equation must be rearranged to isolate the exponential term on one side of the equation and the constant on the other. This gives \((4.7)^x = 135\)

Step 2: Apply the logarithm

Next, apply logarithm to both sides. This can be done with any base, but typically, the common logarithm (base 10) or natural logarithm (base e) is used. For this problem, the natural logarithm (ln) will be used: \(ln((4.7)^x) = ln(135)\)

Step 3: Simplify using logarithm properties

Simplify the left side using the logarithm power rule, which states that \(ln(a^b) = b*ln(a)\). This brings the variable x out of the exponent and yields the following equation: \(x*ln(4.7) = ln(135)\)

Step 4: Isolate x

Rearrange this last equation to solve for x by dividing both sides by \(ln(4.7)\): \(x = \frac{ln(135)}{ln(4.7)}\)

Step 5: Calculate the decimal approximation

Finally, use a calculator to evaluate the right side of the equation and round the result to two decimal places to find the approximate solution to the equation: \(x ≈ 3.76\).

Key Concepts

Logarithms

Logarithms are mathematical operations that help us solve equations involving exponents. Essentially, a logarithm answers the question: "To what power must a specific base be raised, in order to yield a specific number?" For example, in the equation \(b^x = y\), the logarithm base \(b\) of \(y\) is \(x\), which can be expressed as \(\log_b(y) = x\).
Understanding logarithms is crucial, particularly when solving exponential equations as they allow us to "bring down" exponents and work with them algebraically. Two frequently used types of logarithms are the common logarithm (base 10) and the natural logarithm (base \(e\)).
For our problem, logarithms enable us to simplify the exponential equation \((4.7)^x = 135\) into a form that can be easily solved.

Natural Logarithm

The natural logarithm, notated as \(\ln\), is a logarithm with base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. This type of logarithm is particularly useful in natural growth and decay processes, and appears often in calculus and higher-level mathematics.
In our problem, we used the natural logarithm to simplify the equation \((4.7)^x = 135\). By taking \(\ln\) of both sides, we can utilize properties of logarithms to remove the exponent "x," transforming it into \(x \cdot \ln(4.7) = \ln(135)\). This makes our equation linear and much more manageable.
Choosing the natural logarithm or another base, such as the common logarithm, often depends on context or convenience, as they are interchangeable for purposes of solving equations.

Logarithm Properties

Logarithm properties are rules that simplify working with logarithms and are invaluable when manipulating logarithmic expressions. The critical properties used for solving our exponential equation include:

• Power Rule: \(\ln(a^b) = b \cdot \ln(a)\). This rule allows us to take the exponent out in front, making the equation easier to solve when dealing with unknown exponents.
• Product Rule: \(\ln(mn) = \ln(m) + \ln(n)\), useful when dealing with products inside a logarithm.
• Quotient Rule: \(\ln(\frac{m}{n}) = \ln(m) - \ln(n)\), which helps in dealing with divisions inside a logarithm.

In the step-by-step solution, we used the power rule to simplify \(\ln((4.7)^x)\) into \(x \cdot \ln(4.7)\), enabling us to isolate and solve for \(x\).
These properties are foundational to understanding logarithms and solving equations that incorporate them.

Decimal Approximation

Decimal approximation is a technique used to provide a manageable representation of numbers which can be infinitely long. In mathematics, when dealing with log calculations or any irrationals, it's often necessary to approximate these values to make them practically usable.
In solving the exponential equation, after isolating \(x\), we calculate the numerical value of \(x = \frac{\ln(135)}{\ln(4.7)}\). Using a calculator gives us a decimal approximation, \(x \approx 3.76\), rounded to two decimal places. This makes our final answer much easier to interpret and apply.
It’s essential to remember that both scientific and simple calculators can perform these computations, and rounding appropriately to two decimal places provides a balance between precision and practical simplicity.