Problem 84 - Calculus for Biology and Medicine

Question

Solve for \(x\). (a) \(\ln (2 x-3)=0\) (b) \(\log _{2}(1-x)=3\) (c) \(\ln x^{3}-2 \ln x=1\)

Step-by-Step Solution

Answer

(a) \(x = 2\), (b) \(x = -7\), (c) \(x = e\).

Step 1: Understanding the Problem (a)

We need to solve the equation \(\ln(2x - 3) = 0\). The natural logarithm function \(\ln(y)\) equals 0 when \(y = 1\) because \(\ln(1) = 0\). Thus, we set \(2x - 3 = 1\).

Step 2: Solve for x (a)

Starting from the equation \(2x - 3 = 1\), add 3 to both sides: \(2x = 4\). Now, divide both sides by 2 to solve for \(x\): \(x = 2\).

Step 3: Understanding the Problem (b)

We want to solve \(\log_{2}(1-x) = 3\). The logarithm base 2 equals 3 when the argument is \(2^3\). Thus, set \(1-x = 8\).

Step 4: Solve for x (b)

Starting with \(1 - x = 8\), subtract 1 from both sides: \(-x = 7\). Therefore, multiplying both sides by -1 yields \(x = -7\).

Step 5: Simplify the Equation (c)

We have \(\ln x^{3} - 2 \ln x = 1\). Using the power rule, \(\ln x^3 = 3 \ln x\). Substituting into the equation, we have \(3 \ln x - 2 \ln x = 1\), simplifying to \(\ln x = 1\).

Step 6: Solve for x (c)

Since \(\ln x = 1\), raise \(e\) to the power of both sides. Thus, \(x = e^1 = e\). Therefore, \(x = e\).

Key Concepts

Natural Logarithm

Natural logarithm, often denoted as \( \ln \), is a special type of logarithm that has the base of \( e \), where \( e \approx 2.71828 \). This type of logarithm is universally applied in various fields such as mathematics and science.
What makes the natural logarithm unique is its connection to the constant \( e \), which is an irrational number and is extremely significant in mathematical analysis.
In equation (a) of the original exercise, where we have \( \ln(2x - 3) = 0 \), understanding that the natural logarithm of 1 is always 0 simplifies finding the solution. Therefore, if \( \ln(y) = 0 \), then \( y = 1 \). This leads to the equation \( 2x - 3 = 1 \) and solving it gives us \( x = 2 \).

• Natural logarithms relate to the exponential identity \( e^x \).
• If \( \ln(a) = b \), then \( e^b = a \).
• Natural logarithms are essential when modeling continuous growth or decay.

Logarithm Base 2

Logarithms with base 2 are crucial in fields like computer science, because they frequently describe binary operations. Denoted as \( \log_2 \), this logarithm provides insights into binary systems, which are the basis of digital computation.
In part (b) of the original problem, \( \log_2(1-x) = 3 \) implies that the logarithm is determining how many times the base, 2, must be raised to achieve the value on the other side of the equation (i.e., 3). This translates to \( 1-x = 2^3 \), which is \( 8 \). From here, the straightforward arithmetic leads us to \( x = -7 \).

You can remember these key points regarding base 2 logarithms:

• \( \log_2(y) = x \) means \( 2^x = y \).
• They are vital in binary systems and structures.
• Useful for measuring computational complexity and storing binary data efficiently.

Power Rule of Logarithms

The power rule of logarithms is invaluable because it allows simplification of logarithmic expressions involving powers. The rule states that the logarithm of a number raised to a power can be rewritten as the power multiplied by the logarithm of the base number. Mathematically, it’s expressed as \( \ln(x^n) = n \cdot \ln(x) \).
This rule was applied in part (c) of the problem \( \ln x^{3} - 2 \ln x = 1 \). Here, the power rule converts \( \ln x^3 \) into \( 3 \ln x \), making the equation \( 3 \ln x - 2 \ln x = 1\). After simplification, we have \( \ln x = 1 \). By raising \( e \) to both sides, we solve for \( x \), finding \( x = e \).

• The power rule simplifies multiplication within logarithmic expressions.
• It’s handy when solving exponential growth and decay problems.
• Empowers the user to break down complex equations into manageable parts.