Problem 16 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 15-16: Find the number, if possible. (a) \(\log _{8} 1\) (b) \(\log _{9} 9\) (c) \(\log _{5} 0\) (d) \(\log _{6} 6^{7}\) (e) \(5^{\log _{5} 4}\) (f) \(\log _{3} 243\) (g) \(\log _{2} 128\)

Step-by-Step Solution

Answer

(a) 0; (b) 1; (c) Undefined; (d) 7; (e) 4; (f) 5; (g) 7.

Step 1: Understanding the Logarithm Concept

Logarithms help us answer the question: "To what power must the base be raised to obtain a particular number?" When given a logarithmic function like \(\log_{b} a\), it denotes the power to which the base \(b\) must be raised to obtain \(a\).

Step 2: Solving \(\log_8 1\)

Since any number raised to the power of 0 is 1, \(\log_8 1 = 0\). Thus, \(8^0 = 1\).

Step 3: Solving \(\log_9 9\)

The logarithm \(\log_9 9\) equals the power needed for 9 to equal 9, which is 1. Hence, \(\log_9 9 = 1\).

Step 4: Determining \(\log_5 0\)

A logarithm of zero is undefined because no real number exists that, when raised as a power to any base, will result in 0. Therefore, \(\log_5 0\) is undefined.

Step 5: Solving \(\log_6 6^7\)

Applying the logarithmic identity for powers \(\log_b b^x = x\), it follows that \(\log_6 6^7 = 7\).

Step 6: Evaluating \(5^{\log_5 4}\)

Using the property \(b^{\log_b a} = a\), we find \(5^{\log_5 4} = 4\).

Step 7: Finding \(\log_3 243\)

To find the value of \(\log_3 243\), express 243 as a power of 3: \(243 = 3^5\). Thus, \(\log_3 243 = 5\).

Step 8: Evaluating \(\log_2 128\)

Write 128 as a power of 2: \(128 = 2^7\). Hence, \(\log_2 128 = 7\).

Key Concepts

Properties of Logarithms

Logarithms have some fascinating properties that help us solve various types of mathematical problems with ease. One of the primary properties is the power rule for logarithms: If you have a logarithm in the form \( \log_b (a^c) \), it simplifies to \( c \cdot \log_b a \). This is applied when solving problems like \( \log_6 (6^7) \), where the result directly simplifies to 7, because the base and the number are the same.
Another crucial property is the inverse nature of logarithms and exponents. If you see an expression like \( b^{ \log_b a} \), it simplifies directly to \( a \), showcasing how logarithms and exponentials are inverse functions of each other. This principle helps when solving expressions such as \( 5^{ \log_5 4} \), which equals 4, bringing simplicity to what might otherwise seem a complex task.
These properties build the foundation for understanding and working with logarithmic expressions and equations.

Logarithmic Identities

Logarithmic identities serve as important tools in simplifying and evaluating complex logarithm-related expressions. One widely used identity is \( \log_b b = 1 \). It indicates that the logarithm of a base to itself is always 1, as shown when solving \( \log_9 9 \), since 9 raised to the power of 1 is simply 9.
Another identity is \( \log_b 1 = 0 \), based on the notion that any nonzero number raised to the power of 0 results in 1. This identity explains why \( \log_8 1 \) equals 0. These identities allow you to see logarithms not just as isolated expressions but as connected function architectures that streamline calculations.
Mastering these identities is crucial for effective problem-solving in log expressions and broadens analytical thinking.

Exponentiation

Exponentiation involves raising a base number to a specific power, representing multiplication of the base by itself. The expression \( b^n \) signifies the base \( b \) raised to the power of \( n \). This concept is central to understanding logarithms because logarithms essentially identify the exponent.
For example, in \( \log_3 243 \), determining the exponent requires expressing 243 as \( 3^5 \). This is because 3 must be raised to the power of 5 to result in 243, leading to the solution \( \log_3 243 = 5 \).
Grasping how exponentiation works makes evaluating logarithms much more intuitive and highlights the close relationship between these two fundamental mathematical operations.

Undefined Logarithms

An 'undefined logarithm' means that there is no real number that can solve a given logarithm problem. This often happens when trying to take the logarithm of zero or a negative number. For instance, no real number exists such that when a base is raised to that power, the result is zero. Hence, \( \log_5 0 \) is undefined as zero cannot be expressed as a positive base raised to a real number in a real number system.
This concept highlights the importance of domain in logarithmic functions, where acceptable inputs are strictly positive numbers. Thus, understanding undefined logarithms is vital when working with functions and equations, as it ensures that calculations are meaningful and valid.