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Chapter 5: Problem 20

Find each of the following. Do not use a calculator. $$\log 1$$

Short Answer

Expert verified
\(\text{log}(1) = 0\)

Step by step solution

01

- Understand the Logarithm Definition

Recall that \(\text{log}_b(a)\right \) asks the question: To what power must \(\right b\right \) be raised in order to get \(\right a\right \)? Here, \(\text{log}_b(1) = x\right \) translates to \(\right b^x = 1\right \).
02

- Apply the Logarithm Property

Recall that any number raised to the power of 0 is 1, i.e., \(\right b^0 = 1\right \). This implies that \(\right x = 0\right \).
03

- Conclusion

Since we need \(\right \text{log}_b(1)\right \) and \(\right b^x = 1\right \) translates to \(\right x = 0\right \), it follows that \(\right \text{log}_b(1) = 0\right \) for any base \(\right b\right \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

logarithm definition
A logarithm is a mathematical concept that answers the question: To what power must a number, the base \( b \), be raised to produce another number, \( a \)? We represent this relationship as \( \text{log}_b(a) = x \), which can be translated to the exponential form \( b^x = a \). In simpler terms, if you see \( \text{log}_2(8) = 3 \), it means that 2 must be raised to the power of 3 to get 8. This definition is foundational for solving logarithmic problems.
logarithm properties
Logarithms have several useful properties that make calculations easier. Here are some key properties:
  • The Product Property: \( \text{log}_b(MN) = \text{log}_b(M) + \text{log}_b(N) \). This property lets you break down the logarithm of a product into the sum of the logarithms.
  • The Quotient Property: \( \text{log}_b(M/N) = \text{log}_b(M) - \text{log}_b(N) \). This simplifies the logarithm of a quotient into the difference of two logarithms.
  • The Power Property: \( \text{log}_b(M^p) = p \text{log}_b(M) \). Here, a power inside the logarithm can be brought in front as a multiplier.
  • The Zero Property: \( \text{log}_b(1) = 0 \). This crucial property states that the logarithm of 1, for any base \( b \), is always 0 because any number raised to the power of 0 is 1.
These properties help in simplifying and solving logarithmic equations efficiently.
exponentiation
Exponentiation is the process of raising a number, called the base, to the power of an exponent. If you see \( b^x = a \), it means \( b \) is multiplied by itself \( x \) times to equal \( a \). This is the reverse operation of a logarithm. For example, \( 2^3 = 8 \) means 2 is raised to the power of 3, resulting in 8. In logarithms, we're essentially asking what exponent we need to use on the base to get a specific number. Exponentiation helps to connect and transition between exponential and logarithmic forms, making it easier to solve various mathematical problems.

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Most popular questions from this chapter

Solve. $$9^{x-1}=100\left(3^{x}\right)$$

Use a graphing calculator to find the approximate solutions of the equation. $$\log _{5}(x+7)-\log _{5}(2 x-3)=1$$

Use a graphing calculator to find the approximate solutions of the equation. $$5 e^{5 x}+10=3 x+40$$

Consider quadratic functions ( \(a\) )-( h ) that follow. Without graphing them, answer the questions below. a) \(f(x)=2 x^{2}\) b) \(f(x)=-x^{2}\) c) \(f(x)=\frac{1}{4} x^{2}\) d) \(f(x)=-5 x^{2}+3\) e) \(f(x)=\frac{2}{3}(x-1)^{2}-3\) f) \(f(x)=-2(x+3)^{2}+1\) g) \(f(x)=(x-3)^{2}+1\) h) \(f(x)=-4(x+1)^{2}-3\) For which is the line of symmetry \(x=0 ?\)

Solve using any method. $$5^{2 x}-3 \cdot 5^{x}+2=0$$
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