Problem 78 - Calculus for Biology and Medicine

Question

Which real number \(x\) satisfies (a) \(\log _{3} 81=x\) ? (b) \(\log _{5} \frac{1}{25}=x\) ? (c) \(\log _{10} 1000=x ?\)

Step-by-Step Solution

Answer

(a) 4, (b) -2, (c) 3.

Step 1: Understanding Logarithms

A logarithm, expressed as \(\log_{b}(a)\), represents the power to which the base \(b\) must be raised to obtain \(a\). Hence, \(\log_{b}(a)\) means \(b^x = a\) where \(x\) is the answer we seek.

Step 2: Solve \( \log_3 81 \)

To solve \( \log_3 81 = x \), find the power to which 3 must be raised to get 81. Since \(81 = 3^4\), then \(x = 4\). Thus, the solution is \(x = 4\).

Step 3: Solve \( \log_5 \frac{1}{25} \)

For \( \log_5 \frac{1}{25} = x \), express \(\frac{1}{25}\) as \(5^{-2}\). Since \(5^{-2} = \frac{1}{25}\), we deduce that \(x = -2\). So, \(x = -2\).

Step 4: Solve \( \log_{10} 1000 \)

In \( \log_{10} 1000 = x \), identify the power of 10 that results in 1000. We know \(1000 = 10^3\), therefore \(x = 3\). Hence, \(x = 3\).

Key Concepts

Logarithmic Equations

Logarithmic equations are expressions that involve logarithms with an unknown variable. They are essential in solving problems where you need to find the unknown exponent in an exponential equation. Let's break this down in simpler terms. When you see an equation like \( \log_b(a) = x \), it is asking, "To what power should the base \( b \) be raised to get \( a \)?" This equation can also be transformed into its exponential form: \( b^x = a \).

Due to this relationship:

• Logarithmic equations are the inverse operations of exponential equations.
• They allow us to convert multiplication into addition, which is significantly easier to compute and comprehend.
• By solving logarithmic equations, mathematicians can determine the unknown exponent, making these equations quite powerful and useful in various fields.

It's helpful to become comfortable with the rules and properties of logarithms when tackling these equations. Recognizing how to switch between logarithmic and exponential forms is a key skill you'll apply often.

Base and Exponent Relationship

Understanding the base and exponent relationship is vital when dealing with logarithms. The base of the logarithm is the number we repeatedly multiply by itself. The exponent is what specifies how many times this multiplication takes place.

Consider the simple example of \( 2^3 \). Here, '2' is the base, and '3' is the exponent. Thus, \( 2^3 = 2 \times 2 \times 2 = 8 \). Understanding this relationship helps us tackle logarithmic problems because it reveals the connection between multiplication and repeated addition.
Some key points to remember:

• Logarithms undo what exponents do. If you know the result of an exponential expression, using a logarithm with the same base helps you find the exponent.
• The properties of exponents, such as \( a^m \times a^n = a^{m+n} \) and \( (a^m)^n = a^{m \cdot n} \), are invaluable when simplifying logarithmic expressions.
• When you face different bases, it might be necessary to convert them into common bases using properties or even the change of base formula.

Understanding these relationships is not only crucial for solving the exercise at hand but also serves as the foundation for more complex mathematical problems.

Solving Logarithms

Solving logarithmic equations requires identifying the corresponding exponent that satisfies the equation. Let's go through the given examples to better understand.

For \( \log_3 81 = x \), the question is asking what power \( 3 \) needs to be raised to yield \( 81 \). Recognizing that \( 81 = 3^4 \), we can easily determine that \( x = 4 \).

For \( \log_5 \frac{1}{25} = x \), you need to express \( \frac{1}{25} \) as a power of \( 5 \). Since \( \frac{1}{25} = 5^{-2} \), it follows that \( x = -2 \).

Lastly, for \( \log_{10} 1000 = x \), you determine the power to which 10 must be raised to achieve 1000. Recognizing that \( 10^3 = 1000 \), it is clear that \( x = 3 \).

A few tips for solving logarithms effectively:

• Always convert the terms into the same base whenever possible.
• Familiarize yourself with common power relationships, such as \( 10^2 = 100 \), \( 5^3 = 125 \), etc., for quicker computation.
• Use the properties of logarithms, like the power rule or the product rule, to simplify where necessary before solving.

Remember, practice makes perfect, and becoming adept at switching between logarithmic and exponential forms will make these equations feel like second nature.