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Chapter 10: Problem 22

Write in exponential form. $$\log _{\pi} 1=0$$

Short Answer

Expert verified
\( \pi^0 = 1 \)

Step by step solution

01

Understanding the Logarithmic Equation

The given equation is \(\log _{\pi} 1 = 0\). This is a logarithmic equation that needs to be converted into its exponential form.
02

Recall the Logarithm and Exponential Relationship

The general logarithmic form is \(\log_b(a)=c\), which corresponds to the exponential form \((b^c = a)\).
03

Identify 'b', 'c', and 'a' from the Given Equation

In the equation \(\log _{\pi} 1 = 0\), the base \b\ is \pi\, the exponent \c\ is \0\, and \a\ is \1\.
04

Convert to Exponential Form

Using the identified values, rewrite the equation in exponential form: \[\pi^0 = 1\].

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Key Concepts

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

exponential form
Exponential form is a way to express numerical relationships using exponents. It is often used to simplify complex numbers or to express large or small values in a more manageable way.

In an exponential equation, a number called the base is raised to a power, or exponent, to produce another number. The general exponential form is written as: \( b^c = a \).

For example, the equation \(2^3 = 8\) is in exponential form. Here, 2 is the base, 3 is the exponent, and 8 is the result.

Now, let's apply this to our logarithmic equation \(\log_\pi 1 = 0\). When converted to exponential form, this becomes \(\pi^0 = 1\). This shows that raising \(\bp\) to the power of 0 gives us 1.
logarithm properties
Logarithms have several properties that make them useful for solving equations. Here are some key properties:

  • Product Property: \( \log_b(m \cdot n) = \log_b(m) + \log_b(n) \)
  • Quotient Property: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \)
  • Power Property: \( \log_b(m^k) = k \cdot\log_b(m) \)
  • Change of Base Formula: \( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \)


Understanding these properties can make it easier to manipulate logarithmic equations and convert them into exponential form. For example, knowing the Power Property helps us understand that \(\log_\pi 1 = 0\) implies \(\pi^0 = 1\).
base and exponent
In both exponential and logarithmic equations, the base and exponent are crucial elements.

  • The base is the number that is being raised to a power.
  • The exponent (or power) tells you how many times to multiply the base by itself.


In the equation \ \log_\pi 1 = 0 \, \pi is the base and 0 is the exponent.

When converting from logarithmic to exponential form, understanding the base and exponent helps us rewrite the equation correctly. For instance, in our case, \(\log_\pi 1 = 0\) converts to \(\pi^0 = 1\). This means that any value raised to the power of zero is always 1.

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Most popular questions from this chapter

Use the special properties of logarithms to evaluate each expression. $$\log _{9} \sqrt[3]{9}$$

Use the change-of-base rule (with either common or natural logarithms) to approximate each logarithm to four decimal places. $$ \log _{19} 0.8325 $$

Solve each equation. Give exact solutions. $$ \log _{2} x+\log _{2}(x-6)=4 $$

Use the change-of-base rule (with either common or natural logarithms) to approximate each logarithm to four decimal places. $$ \log _{3} \sqrt{2} $$

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. See Example 5. $$ \log _{10} 18 $$
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