Problem 77 - Calculus for Biology and Medicine

Question

Which real number \(x\) satisfies (a) \(\log _{1 / 2} 32=x ?\) (b) \(\log _{1 / 3} 81=x ?\) (c) \(\log _{10} 0.001=x ?\)

Step-by-Step Solution

Answer

(a) \(x = -5\), (b) \(x = -4\), (c) \(x = -3\).

Step 1: Understanding the Logarithm Definition

Remember that the logarithm \( \log_b a = c \) means \( b^c = a \). Thus, we need to express the exponential equation corresponding to each of the given logs.

Step (a) Compute \(\log _{1 / 2} 32=x\)

Convert \( \log_{1/2} 32 = x \) into the exponential form: \( (1/2)^x = 32 \. \). Recognize that 32 can be expressed as \( 2^5 \), so \( (1/2)^x = 2^{-5} \, \) leading to \( x = -5 \).

Step (b) Compute \(\log _{1 / 3} 81=x\)

Convert \( \log_{1/3} 81 = x \) to the exponential form: \( (1/3)^x = 81 \). Recognize that 81 can be expressed as \( 3^4 \, \), hence \( (1/3)^x = 3^{-4} \, \) giving us \( x = -4 \).

Step (c) Compute \(\log_{10} 0.001=x\)

Write the equation in exponential form: \( 10^x = 0.001 \). Recall that \( 0.001 = 10^{-3} \), so \( 10^x = 10^{-3} \, \) which results in \( x = -3 \).

Step 5: Conclusion of Solution Steps

The solutions for each part are (a) \( x = -5 \), (b) \( x = -4 \), and (c) \( x = -3 \).

Key Concepts

Understanding Real Numbers

Real numbers are all the numbers that you encounter on the number line. This includes both rational numbers (like fractions and whole numbers) and irrational numbers (like the square root of 2 or pi). Real numbers can be positive, negative, or zero, and they are useful for expressing almost any quantity we come across in the real world.
In mathematical terms, real numbers include:

• Integers: These are whole numbers that can be positive, negative, or zero. Examples include -2, 0, 7.
• Fractions: These represent divisions of integers, like 1/2, 3/4, and 5/1.
• Irrational numbers: Numbers that can't be expressed as a simple fraction. Common examples are \( \) and .
• Decimal numbers: Able to be finite, like 0.75, or repeating, like 0.333....

While working with logarithms, you often solve equations that require you to find real number solutions. This helps us express results in a way that's meaningful for problems involving growth, decay, or scaling.

Exponential Equations and Their Role

Exponential equations are equations in which variables appear as exponents. They are an essential part of understanding how things grow (or shrink) exponentially under a constant rate of change. They look something like this: \(b^x = a\).
Here's how exponential equations work:

• Base (\(b\)): The constant number that is raised to the power of \(x\).
• Exponent (\(x\)): The unknown that determines the result when the base is raised to this power.
• Result (\(a\)): The product or outcome of the exponentiation.

When you solve exponential equations using logarithms, you employ the property that logarithms are the inverse operation of taking powers, making it easier to isolate the variable. For example, to solve \(b^x = a\), you would take the logarithm of both sides to isolate \(x\).
This is particularly helpful when the equation involves large numbers or fractions, as logarithms allow handling them in a simpler and more manageable way.

Decoding Mathematical Notation

Mathematical notation acts as a universal language for expressing mathematical ideas precisely and succinctly. In the context of logarithms, several symbols and notations are essential to master:

• Logarithm: Expressed as \(log_b a = c\), it means you're finding the power you must raise \(b\) to get \(a\).
• Exponent: Indicated usually as a superscript, it shows how many times the base is used as a factor, like \(b^c = a\).
• Base: Represents the number that is multiplied by itself when raised to an exponent, crucial for solving logarithmic equations.

Understanding notation helps demystify complex expressions into clear, solvable segments. For logarithmic calculations, familiarity with the notation ensures you can read equations accurately and apply the right techniques to find solutions. This translates into an ability to handle a wide variety of mathematical problems efficiently.