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Chapter 12: Problem 59

Solve each equation. \(\log _{\pi} \pi^{4}=x\)

Short Answer

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Step by step solution

01

Understand the Logarithmic Equation

The given equation is \(\text{log}_{\text{base} \ \pi}(\text{\ \pi^{4}})=x\). In logarithmic form, it represents the exponent to which the base must be raised to produce the number inside the logarithm.
02

Apply the Logarithmic Identity

Recall the logarithmic identity \(\text{log}_{b}(b^{y})=y\). For any logarithm \(log_{b}(b^{y})=y\), the answer is simply \(\text{y}\) since \( \log_{b}\) and the exponent \( \text{}^{y} \) are inverses.
03

Identify Base and Exponent

Here, the base \(\ \ \pi\ \) and the exponent is \(4\), so using \( \log_{\text{base} }(\text{base}^{y})=y \), we can simplify:
04

Solve the Equation

Simplify \( \log_{\text{base} \pi}(\ \pi^{4} \) = 4. Therefore, \(x = 4\).

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Key Concepts

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

logarithmic identities
Logarithmic identities make solving equations easier. For the given problem, the identity \[ \log_{b}(b^{y})=y \] is used.
This identity tells us that if the base of the logarithm and the base of the exponent are the same, the result is simply the exponent.

For example: \[ \log_{2}(2^{3})=3, \log_{10}(10^{5})=5 \]

The given problem, \[ \log_{\pi}(\pi^{4})=x \, \] is solved by recognizing the matching bases and applying the identity.

Since \[\log_{\pi}(\pi^{4})=4,\] it's clear that \( x \) equals 4.
base of logarithms
To understand logarithms, knowing the base is crucial. The base of a logarithm indicates the number that is raised to a power. For instance, in \(\log_{2}(8)\), the base is 2.

Common bases you might encounter include:
  • 2 (binary logarithms, used in computing)
  • 10 (common logarithms, often used in science)
  • \( e \) (natural logarithms, used in calculus)
In the exercise, the base is \( \pi \, \, (\approx 3.14) \,\), which is a unique base used for this problem.
Understanding the base helps in simplifying and solving logarithmic equations using identities like the one we applied.
exponential functions
Exponential functions have the form \( f(x) = b^{x} \). They are inverses of logarithmic functions. In exponential functions, the base \( b \) is raised to the power of \( x \).

Here are some key properties of exponential functions:
  • The base \( b \) is a constant and must be positive and not equal to 1.
  • The function \( f(x) = b^{x} \) grows rapidly for \( b > 1 \).
  • When \( 0 < b < 1 \), the function decreases and approaches zero.
For the exercise, we consider the function \( \pi^{4} \). Given \( \log_{b}(y) \), it asks for the power that \( b \) must be raised to, to result in \( y \).

In this case, solving for \( x \) involves understanding that if \( y = b^{x} \), then \( x = \log_{b}(y) \).

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Most popular questions from this chapter

Solve each equation. \(\log _{4}(2 x+4)=3\)

Solve each equation. \(\log _{4} \sqrt{64}=x\)

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. $$\log _{7} \frac{\sqrt[3]{13}}{p q^{2}}$$

Use the special properties of logarithms to evaluate each expression. \(\log _{5} 1\)

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