Question

What are the values of the following logarithms? \((a) \log _{10} 10,000\) \((c) \log _{3} 81\) (b) \(\log _{10} 0.0001\) \((d) \log _{5} 3,125\)

Short answer

(a) 4, (b) -4, (c) 4, (d) 5.

Step-by-step solution

Analyzing Logarithms

A logarithm asks the question: 'To what power must the base be raised, to obtain a certain number?' For example, with \(\log_b x = y\), we need to find \(y\) such that \(b^y = x\).

Solve (a) \(\log_{10} 10,000\)

Since the base is 10 and the logarithm asks what power 10 must be raised to produce 10,000, rewrite 10,000 as a power of 10: \(10^4 = 10,000\). Thus, \(\log_{10} 10,000 = 4\).

Solve (b) \(\log_{10} 0.0001\)

To find this logarithm, identify 0.0001 as a power of 10: \((0.0001 = 10^{-4})\). Hence, \(\log_{10} 0.0001 = -4\).

Solve (c) \(\log_{3} 81\)

Rewrite 81 as a power of 3: \(3^4 = 81\). Therefore, \(\log_{3} 81 = 4\).

Solve (d) \(\log_{5} 3,125\)

Rewrite 3,125 as a power of 5: \(5^5 = 3,125\). Thus, \(\log_{5} 3,125 = 5\).

Key concepts

Exponents

Exponents are a way to express a number multiplied by itself a certain number of times. For instance, when we see something like \( 10^4 \), it means \(10\) is being multiplied by itself four times (i.e., \( 10 \times 10 \times 10 \times 10 \)).

This is a powerful way to represent large numbers in a compact form. Exponents have some straightforward rules that make them easy to work with:

• Multiplication Rule: When multiplying numbers with the same base, add the exponents. So, \( a^m \times a^n = a^{m+n} \).
• Division Rule: When dividing, subtract the exponents. Hence, \( a^m / a^n = a^{m-n} \).
• Power Rule: To raise a power to another power, multiply the exponents: \( (a^m)^n = a^{m \times n} \).

Understanding exponents makes it easier to grasp logarithms since logarithms are essentially the opposite operation of exponentiation.

Base of Logarithm

The base of a logarithm is key to understanding logarithmic expressions. It tells us which number we're repeatedly multiplying to achieve another number.

Consider the expression \( \log_b x = y \). Here, "\( b \)" is the "base". The question asked by this expression is: "To what power must 'b' be raised to get 'x'?"

It's important to note that:

• Common Bases: The base 10 logarithm is often called the common logarithm and is very frequently used, especially in scientific contexts.
• Natural Bases: The base of 'e' (approximately 2.718) is known as the natural logarithm and appears frequently in calculus and continuous growth calculations.

Always remember that the base is crucial to finding the right power or exponent in logarithmic equations.

Mathematical Problem Solving

Mathematical problem solving is a structured approach to tackle math problems effectively. To solve logarithmic problems, applying a consistent methodology helps make complex questions more approachable.

Here are steps you can take for problem solving with logarithms:

• Understand the Question: The first step is always to understand what the logarithm expression is asking.
• Rewrite the Problem: Convert logarithmic expressions into exponential form if needed, as seen in \( \log_b x = y \rightarrow b^y = x \).
• Calculate Step by Step: Break the problem into smaller steps, especially when dealing with complex numbers.
• Verify Your Solution: Always substitute your answer back into the original equation to verify its correctness.

Following these steps can make solving logarithmic problems less daunting and more intuitive.

Power of a Number

The power of a number signifies how many times to use the number in a multiplication.

For example, in \(5^3\), 5 is the base, and 3 is the exponent or power, indicating that 5 is multiplied by itself three times. This concept is central to both exponents and logarithms.

Some facts about powers can be very helpful:

• Zero Power: Any number raised to the power of zero is always 1, like \( a^0 = 1 \).
• Negative Power: A negative exponent means "take the reciprocal and then apply the positive exponent": \( a^{-n} = \frac{1}{a^n} \).

Understanding the power of a number is essential when rewriting numbers into different forms to solve logarithms or exponents efficiently.