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Chapter 12: Problem 15

Write in logarithmic form. \(5^{0}=1\)

Short Answer

Expert verified
\(\log_5(1) = 0\)

Step by step solution

01

- Understand the Exponential Form

Recall that the given equation is in the form of an exponent: \[5^0 = 1\]
02

- Know the Logarithmic Form

Understand that logarithmic form is written as: \[\log_b(a) = c\] where \(b^c = a\)
03

- Apply the Logarithmic Form

Identify the base, exponent, and result in the given exponential equation. In this case, the base \(b\) is 5, the exponent \(c\) is 0, and the result \(a\) is 1. Rewrite it in logarithmic form as: \[\log_5(1) = 0\]

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Key Concepts

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

Introduction to Exponential Form
When we talk about exponential form, we are dealing with expressions where a number, known as the base, is raised to a power called the exponent. These expressions look like this: \(b^c = a\). For example, in the equation \(5^0 = 1\), 5 is the base, 0 is the exponent, and 1 is the result.
Exponents tell us how many times to multiply the base by itself. An exponent of 0 is a special case. No matter what the base is, any number to the power of 0 equals 1.
If you have \(x^0\), for any value of x (except 0), it always equals 1.
This is a fundamental concept to understand before moving on to logarithms.
Understanding Logarithmic Form
The logarithmic form is another way to represent exponential relationships. It essentially answers the question: 'To what exponent must we raise the base to get a certain number?'. The general logarithmic form is \[ \log_b(a) = c \] where \(b\) is the base, \(a\) is the number you're looking at, and \(c\) is the exponent.
In simpler terms, the logarithmic form takes the result of an exponentiation and tells you what power you raised the base to achieve that result.
For instance, using our previous example \(5^0 = 1\), the log form of this would ask: 'To what power must we raise 5 to get 1?' The answer is 0, and so we write it as: \[ \log_5(1) = 0 \]
Delving into Exponents
Exponents are a shorthand way to express repeated multiplication of the same number by itself. They are crucial in many areas of math and science. For example:
\(2^3\) means 2 multiplied by itself 3 times, or \(2 \times 2 \times 2 = 8\).
Exponents follow specific rules, some of which include:
  • Any number to the power of 1 is the number itself. For instance, \(5^1 = 5\)
  • Any number to the power of 0 is 1, except 0 (i.e., \(x^0 = 1\) for \(x eq 0\))
  • Multiplying like bases means you add the exponents: \((a^m) \times (a^n) = a^{m+n}\)
  • Dividing like bases means you subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\)
  • Power of a power means you multiply the exponents: \((a^m)^n = a^{m \times n}\)
Getting comfortable with these rules can make working with exponents much more manageable.
Remember, these rules are tools that help you manipulate and solve equations involving exponents more effectively.

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Most popular questions from this chapter

Use the properties of logarithms to write each expression as a single logarithm. Assume that all variables are defined in such a way that the variable expressions are positive, and bases are positive numbers not equal to 1. $$\left(\log _{a} r-\log _{a} s\right)+3 \log _{a} t$$

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$\log _{10} 2=0.3010 \text { and } \log _{10} 9=0.9542$$ Use these values and the properties of logarithms to evaluate each expression. DO NOT USE A CALCULATOR. $$\log _{10} 18$$

Solve each equation. \(\log _{x} 5=\frac{1}{2}\)

Use the properties of logarithms to express each logarithm as a sum or difference of logarithms, or as a single logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{7}(4 \cdot 5)$$

$$ \left(\frac{4}{3}\right)^{x}=\frac{27}{64} $$
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