Problem 78 - Calculus for Biology and Medicine

Question

Which real number \(x\) satisfies (a) \(\log _{4} 64=x ?\) (b) \(\log _{1 / 5} 625=x ?\) (c) \(\log _{10} 10,000=x\) ?

Step-by-Step Solution

Answer

(a) 3, (b) -4, (c) 4.

Step 1: Understand the Concept

The logarithmic expression \( \log_b a = x \) means that \( b^x = a \). We need to find \( x \) such that when the base is raised to \( x \), it equals the argument.

Step 2: Solve Part (a)

Given \( \log_{4} 64 = x \), we rewrite it as \( 4^x = 64 \). Since \( 64 = 4^3 \), we equate \( 4^x = 4^3 \). This implies \( x = 3 \).

Step 3: Solve Part (b)

Given \( \log_{1/5} 625 = x \), we rewrite it as \( (1/5)^x = 625 \). Recognize \( 625 = 5^4 \). Thus, \( (5^{-1})^x = 5^4 \) which implies \( 5^{-x} = 5^4 \). So, \( -x = 4 \) and therefore \( x = -4 \).

Step 4: Solve Part (c)

Given \( \log_{10} 10,000 = x \), rewrite it as \( 10^x = 10,000 \). Recognize \( 10,000 = 10^4 \). So, \( 10^x = 10^4 \) and hence, \( x = 4 \).

Key Concepts

Real Numbers

Real numbers are numbers that encompass the broad range of values used in everyday mathematics. They include various kinds of numbers like integers, rational numbers, and irrational numbers.

• **Integers**: These are whole numbers that can be positive, negative, or zero. Examples include -2, 0, 1, 2, etc.
• **Rational Numbers**: These are numbers that can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \eq 0 \). A simple example is \( \frac{1}{2} \).
• **Irrational Numbers**: These cannot be written as simple fractions. Examples include \( \sqrt{2} \) and \( \pi \).

Real numbers are used in numerous mathematical calculations and are essential for solving equations involving logarithms, like in our exercise. Here, \( x \) represents a real number we need to find by using the properties of logarithms.

Logarithmic Expressions

Logarithmic expressions give us insight into how many times we need to use one number in multiplication to get another. The basic form is \( \log_b a = x \, \) which means we must multiply \( b \) (the base) by itself \( x \) times to reach the number \( a \).

• **Base**: The number that is repeatedly multiplied. In \( \log_4(64) \, \) the base is \( 4 \).
• **Argument**: The resulting number after multiplying. In \( \log_4(64) \, \) the argument is \( 64 \).
• **Value of x**: This is what we are solving for, which tells us the number of times the base has to be multiplied to get the argument.

Logarithmic expressions can often be rewritten in exponential form, which makes solving them more direct as shown in our step-by-step approach.

Exponential Equations

Exponential equations involve expressions with exponents, where a number (base) is raised to a power. In logarithms, understanding these equations helps identify the value of \( x \).

An exponential equation might look like \( b^x = a \. \) Let’s break it down:

• In the equation \( 4^x = 64 \, \) \( 4 \) is the base and \( 64 \) is the result when \( 4 \) is raised to power \( x \). \( x \) is found by identifying that \( 64 = 4^3 \), therefore, \( x = 3 \).
• For \( (1/5)^x = 625 \, \) rewriting \( 625 \) as \( 5^4 \, \) and considering \( (1/5) \) as \( 5^{-1} \, \) we equate \( (-x) \) to \( 4 \) yielding \( x = -4 \).
• When \( 10^x = 10,000 \, \) realizing \( 10,000 = 10^4 \) directly implies \( x = 4 \).

Exponential equations are highly useful in various fields including sciences and economics, aiding with growth calculations and decay models.