Problem 52 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 51-52: Approximate \(x\) to three significant figures. (a) \(\log x=1.8965\) (b) \(\log x=4.9680\) (c) \(\log x=-2.2118\) (d) \(\ln x=3.7\) (e) \(\ln x=0.95\) (f) \(\ln x=-5\)

Step-by-Step Solution

Answer

(a) 78.7, (b) 9.29×10^4, (c) 0.00614, (d) 40.4, (e) 2.59, (f) 0.00674

Step 1: Definition of Logarithmic Functions

Understand what it means when given \( \log x \) or \( \ln x \). \( \log x \) refers to the common logarithm based on 10 and \( \ln x \) refers to the natural logarithm based on \( e \). To find \( x \), we need to find powers of 10 or \( e \) respectively. Let's proceed accordingly.

Step 2: Solve for Part (a)

Given \( \log x = 1.8965 \), use the property of logarithms to find \( x \):\[x = 10^{1.8965}\]Calculating the expression, \( x \approx 78.7 \). Hence, \( x \) to three significant figures is \( 78.7 \).

Step 3: Solve for Part (b)

Given \( \log x = 4.9680 \), we find \( x \) by: \[x = 10^{4.9680} \]Compute the value, resulting in \( x \approx 9.288\times 10^4 \). The answer to three significant figures is \( 9.29 \times 10^4 \).

Step 4: Solve for Part (c)

Given \( \log x = -2.2118 \), compute \( x \) with: \[x = 10^{-2.2118} \]This results in \( x \approx 0.006144 \). To three significant figures, \( x \approx 0.00614 \).

Step 5: Solve for Part (d)

Given \( \ln x = 3.7 \), find \( x \) using the exponential function:\[x = e^{3.7}\]This calculation gives \( x \approx 40.4 \). Therefore, \( x \approx 40.4 \) to three significant figures.

Step 6: Solve for Part (e)

Given \( \ln x = 0.95 \), compute \( x \):\[x = e^{0.95} \]As a result, \( x \approx 2.585 \). Therefore, \( x \approx 2.59 \) when rounded to three significant figures.

Step 7: Solve for Part (f)

Given \( \ln x = -5 \), solve for \( x \):\[x = e^{-5}\]This results in \( x \approx 0.006738 \). To three significant figures, \( x \approx 0.00674 \).

Key Concepts

Common Logarithm

The common logarithm, denoted as \(\log x\), is based on base 10. This means that when you see \(\log x\), you should interpret it as \(\log_{10} x\). In simpler terms, it's the power to which you need to raise 10 to get \(x\). For instance, if \(\log x = 2\), then \(x = 10^2 = 100\). The common logarithm is often used in scientific fields and real-world applications where the decimal system is prevalent. Here's why understanding common logarithms is crucial:

• They simplify complex multiplication and division into addition and subtraction, making calculations easier.
• They help in computing large numbers seen in probability and statistics as they reduce the scale.

When solving logarithmic equations involving common logarithms, like in the exercise where \(\log x\) values are given, the goal is to express \(x\) as a power of 10. This allows us to find the numerical value of \(x\) succinctly.

Natural Logarithm

The natural logarithm, expressed as \(\ln x\), uses the base \(e\), an irrational number approximately equal to 2.71828. It is especially significant in calculus and areas focused on continuous growth processes.The natural logarithm answers the question: "To what power must \(e\) be raised, to produce \(x\)?" For example, if \(\ln x = 1\), this implies \(x = e^1 \approx 2.718\).It comes in handy because:

• It simplifies calculations involving exponential growth and decay, such as population dynamics and radioactive decay.
• It's fundamental in calculus, especially in solving differential equations and integration.

In exercises involving \(\ln x\), like the one given, your task is to determine \(x\) by raising \(e\) to the power given by the logarithm, hence accomplishing the natural transformation of the logarithmic equation.

Exponential Function

An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. In the context of common and natural logarithms, the exponential functions of interest are \(10^x\) and \(e^x\), respectivelyLet's break down why exponential functions are important:

• They model real-world phenomena whose rate of change is proportional to the current value, such as compound interest and population growth.
• They enable the reversal of logarithmic functions, allowing us to solve equations that otherwise seem complex.

While working with assignments such as logarithmic problems, exponential functions are your go-to tool for finding \(x\) from known logarithm values. They transform logarithmic equations back to their original form, providing a clear path to a numerical solution. For example, by using \(x = 10^{1.8965}\), you find \(x\) easily through the power raised to base 10.