Problem 10 - Contemporary Precalculus

Question

Solve the equation. First express your answer in terms of natural logarithms (for instance, \(x=(2+\ln 5) /(\ln 3)) .\) Then use a calculator to find an approximation for the answer. $$2^{x}=9$$

Step-by-Step Solution

Answer

Question: Solve the equation \(2^{x}=9\) and express the answer in terms of natural logarithms. Then, approximate the numerical value of x. Answer: In terms of natural logarithms, the solution is \(x=\frac{\ln(9)}{\ln(2)}\). The approximate numerical value of x is 3.17.

Step 1: Take the natural logarithm of both sides of the equation

To solve for x, we will take the natural logarithm of both sides of the equation: \(\ln(2^x)=\ln(9)\).

Step 2: Use the properties of logarithms to simplify

By using the power rule of logarithms, which states that \(\ln(a^b)=b\ln(a)\), we can simplify the left side of the equation: \(x\ln(2)=\ln(9)\).

Step 3: Isolate x

To isolate x, we will divide both sides of the equation by \(\ln(2)\): \(x=\frac{\ln(9)}{\ln(2)}\). This is the solution expressed in terms of natural logarithms.

Step 4: Approximate the solution

Use a calculator to approximate the numerical value of x: \(x\approx\frac{\ln(9)}{\ln(2)}\approx3.17\). The approximate solution for x is 3.17.

Key Concepts

Exponential Equations

Exponential equations are equations in which variables occur as exponents. These types of equations may seem daunting at first. However, they're very manageable once you understand the techniques for solving them.
When you see an equation like \(2^x = 9\), the goal is to find the value of the variable \(x\) that makes the equation hold true.

• To solve exponential equations, we often use logarithms, as they serve to bring the exponent down to where we can manipulate it more easily.
• For example, taking the natural logarithm (often denoted as \(\ln\)) of both sides transforms the equation into a form that allows for solving the exponent.
• This process converts the problem from an exponential equation into a linear one, making it much easier to work with.

Mastering exponential equations gives you a powerful tool to tackle a wide range of real-world and mathematical problems.

Natural Logarithms

Natural logarithms are a special type of logarithm with the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. Natural logarithms are represented by \(\ln(x)\), and they play a crucial role in solving exponential equations.
When dealing with problems like \(2^x = 9\), using natural logarithms is especially helpful:

• They allow us to convert the problem into a manageable algebraic form, \(x\ln(2) = \ln(9)\).
• The properties of logarithms simplify this conversion, making computations straightforward and systematic.
• Natural logarithms are commonly used in calculus, physics, and other scientific domains because many natural processes grow exponentially.

Understanding natural logarithms equips you with the tools needed for further studies in mathematics and science.

Logarithmic Properties

Logarithmic properties are essential tools that make it easier to manipulate logarithmic expressions. One key property used in solving exponential equations is the power rule. The power rule states that \(\ln(a^b) = b\ln(a)\), allowing us to bring exponents down to the front of a log expression, just like we did with \(2^x = 9\).
Here's a breakdown of some useful logarithmic properties:

• Product Rule: \(\ln(ab) = \ln(a) + \ln(b)\). This property helps when you multiply within a log.
• Quotient Rule: \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\). Use this when you divide within a log.
• Change of Base Formula: \(\log_b(a) = \frac{\ln(a)}{\ln(b)}\). It allows you to convert logarithms to different bases, which can be quite useful depending on the situation.

Knowing these properties can greatly simplify your work with logs and help solve equations across different contexts. They are fundamental in algebra and higher-level mathematics, serving as a bridge between different types of mathematical problems.