Problem 58
Question
Let \(b\) be any positive real number such that \(b \neq 1\) . What must log, 1 be equal to? Verify the result.
Step-by-Step Solution
Verified Answer
\(\log_b{1} = 0\) for any positive real number \(b\) except \(b = 1\).
1Step 1: Understand the Logarithm Definition
The logarithm log base \(b\) of \(x\), written as \(\log_b{x}\), is defined as the power \(y\) to which the base \(b\) must be raised to obtain \(x\). In other words, \(b^y = x\).
2Step 2: Apply the Definition to the Problem
We need to find \(\log_b{1}\). Using the definition of logarithm, \(b^y = 1\). Here, we'll solve to find what \(y\) must be.
3Step 3: Solve the Equation
Since any non-zero number to the power of 0 is 1, \(y = 0\). Hence, \(b^0 = 1\).
4Step 4: Verify the Result
We established that \(b^0 = 1\), and this holds true for any positive real number \(b\) except \(b = 0\), which confirms that \(\log_b{1} = 0\) for \(b eq 1\).
Key Concepts
Logarithmic DefinitionPower of ZeroBase of Logarithm
Logarithmic Definition
Logarithms connect multiplication and exponentiation in a unique and useful way. They answer the question: "To what power must the base be raised to obtain a certain number?".
- Given a logarithm expressed as \( \log_b{x} \), where \(b\) is the base and \(x\) is the number we're interested in, we solve for a power \(y\) such that \(b^y = x\).
- For example, if you have \( \log_2{8} \), you're seeking the power \(y\) for which \(2^y = 8\). In this case, \(y = 3\) because \(2^3 = 8\).
Understanding this definition is crucial, as all logarithmic identities and rules are derived from it. It serves both mathematical and practical purposes, such as simplifying complex multiplication into manageable addition through the use of log tables and calculators.
- Given a logarithm expressed as \( \log_b{x} \), where \(b\) is the base and \(x\) is the number we're interested in, we solve for a power \(y\) such that \(b^y = x\).
- For example, if you have \( \log_2{8} \), you're seeking the power \(y\) for which \(2^y = 8\). In this case, \(y = 3\) because \(2^3 = 8\).
Understanding this definition is crucial, as all logarithmic identities and rules are derived from it. It serves both mathematical and practical purposes, such as simplifying complex multiplication into manageable addition through the use of log tables and calculators.
Power of Zero
The concept of raising a number to the power of zero is a fundamental aspect of mathematics. It is often one of the first rules taught when learning about exponents. Here's why this is important:
- Any non-zero number raised to the power of zero equals 1. This crucial rule is symbolized as \(b^0 = 1\), regardless of the specific number chosen for \(b\).
- This is not just a random occurrence but follows logically from the properties of exponents. Starting with \( b^1 = b \), as you decrease the power by 1 (subtracting 1 from \(b^1\)), you will divide the result by \(b\), leading to \(b^0 = 1\).
Understanding this concept helps explain why \( \log_b{1} = 0 \). Recalling the logarithmic definition, the power we must raise \(b\) to reach 1 is indeed 0.
- Any non-zero number raised to the power of zero equals 1. This crucial rule is symbolized as \(b^0 = 1\), regardless of the specific number chosen for \(b\).
- This is not just a random occurrence but follows logically from the properties of exponents. Starting with \( b^1 = b \), as you decrease the power by 1 (subtracting 1 from \(b^1\)), you will divide the result by \(b\), leading to \(b^0 = 1\).
Understanding this concept helps explain why \( \log_b{1} = 0 \). Recalling the logarithmic definition, the power we must raise \(b\) to reach 1 is indeed 0.
Base of Logarithm
The base of a logarithm is an integral part of understanding logarithms. It fundamentally determines how the logarithmic function behaves:
- The base \(b\) of a logarithm \(\log_b{x}\) must be a positive real number, so \(b > 0\) and \(b eq 1\).
- The reason \(b\) cannot be 1 is that any power of 1 still equals 1, making the logarithm undefined or indeterminate.
When solving logarithmic equations or interpreting logarithmic scales, recognizing the base is essential. The base controls the rate of growth in the corresponding exponential function. For example, with \(\log_2{x}\), we are continuously doubling as we increase \(x\). Each different base gives the logarithmic function a unique characteristic, allowing its application in diverse fields from biology to computer science.
- The base \(b\) of a logarithm \(\log_b{x}\) must be a positive real number, so \(b > 0\) and \(b eq 1\).
- The reason \(b\) cannot be 1 is that any power of 1 still equals 1, making the logarithm undefined or indeterminate.
When solving logarithmic equations or interpreting logarithmic scales, recognizing the base is essential. The base controls the rate of growth in the corresponding exponential function. For example, with \(\log_2{x}\), we are continuously doubling as we increase \(x\). Each different base gives the logarithmic function a unique characteristic, allowing its application in diverse fields from biology to computer science.
Other exercises in this chapter
Problem 57
For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth. $$\log (\sqrt{2})$$
View solution Problem 57
Explore and discuss the graphs of \(f(x)=\log _{1}(x)\) and \(g(x)=-\log _{2}(x) .\) Make a conjecture based on the result.
View solution Problem 58
Recall that an exponential function is any equation written in the form \(f(x)=a \cdot b^{x}\) such that \(a\) and \(b\) are positive numbers and \(b \neq 1\) .
View solution Problem 58
For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth. $$\ln (\sqrt{2})$$
View solution