Problem 2 - College Algebra

Question

Solve each exponential equation in Exercises \(1-26\) Express the solution set in terms of natural logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$10^{x}=8.07$$

Step-by-Step Solution

Answer

The solution to the equation \(10^{x}=8.07\) is \(x ≈ 0.91\).

Step 1: Set up the equation

Given the equation to solve: \(10^{x}=8.07\).

Step 2: Apply the natural logarithm

Apply the natural logarithm (ln) to both sides of the equation: \(\ln(10^{x}) = \ln(8.07)\). The natural logarithm of 10 to the power of x can be rewritten using the properties of logarithms. This results in the equation: \(x \cdot \ln(10) = \ln(8.07)\).

Step 3: Solve for x

Solve the equation for x by dividing both sides of the equation by \(\ln(10)\): \(x = \frac{\ln(8.07)}{\ln(10)}\).

Step 4: Get the decimal approximation

Now, use a calculator to find the decimal approximation of x: \(x ≈ 0.91\).

Key Concepts

Understanding Natural Logarithms

Natural logarithms are essentially logarithms with a base of Euler's number, denoted as 'e', where 'e' is approximately equal to 2.71828. It's a key mathematical constant used in many areas of calculus and complex mathematics. The natural logarithm of a number 'x' is written as \( \ln(x) \), and it answers the question: 'To what power must we raise e to obtain the number x?'

For instance, when working on exponential equations like \(10^{x}=8.07\), converting to natural logarithms can simplify the process. By applying \(\ln\) to both sides as in \(\ln(10^{x}) = \ln(8.07)\), we utilize the inverse relationship between logarithms and exponentiation, allowing us to isolate the variable 'x' and solve the equation.

Properties of Logarithms in Solving Equations

Logarithms have properties that make solving exponential equations more manageable. One of these properties is the power rule, which states that \(\ln(a^{n}) = n \cdot \ln(a)\). This property is used when solving equations like \(10^{x}=8.07\) by transforming \(\ln(10^{x})\) into \(x \cdot \ln(10)\), as seen in the step by step solution. Other useful properties include the product rule, \(\ln(ab) = \ln(a) + \ln(b)\), and the quotient rule, \(\ln(\frac{a}{b}) = \ln(a) - \ln(b)\). These properties allow us to manipulate the equation into a form where 'x' can be isolated and solved more conveniently.

Decimal Approximation for Solutions

Often, solutions to logarithmic equations are irrational numbers that can't be expressed as exact decimals. Therefore, we use a decimal approximation to represent the solution in a more understandable and practical way. A calculator can be used to compute this approximation, yielding a rounded value. For example, the solution to an equation like \(10^{x}=8.07\) is first expressed with natural logarithms as \(x = \frac{\ln(8.07)}{\ln(10)}\), then approximated to two decimal places using a calculator, giving us \(x \approx 0.91\). In applied mathematics and sciences, decimal approximations are crucial, as they allow us to translate complex numbers into usable figures with a specified level of accuracy.