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Chapter 3: Problem 27

Use the properties of logarithms to simplify the expression. $$\log _{\pi} \pi$$

Short Answer

Expert verified
The simplified expression is 1.

Step by step solution

01

Identify the given log expression

The given expression is \( \log _{\pi} \pi \). This is a logarithm where the base (\( \pi \), which is a constant used in mathematical equations that represents the ratio of a circle鈥檚 circumference to its diameter) is equal to the argument (the number to which the base is raised to produce the logarithm).
02

Apply the logarithm property

There is a property of logarithms that states: for any base \(b\) (where \(b \neq 1\) and \(b > 0\)), \( \log_b{b} = 1 \). This rule essentially stipulates that when the argument and the base of the logarithm are the same, the result is 1. Apply this property to simplify the given logarithm expression.
03

Simplify the given log expression

The simplified value of the logarithm \( \log _{\pi} \pi \) based on the applied property is 1.

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Most popular questions from this chapter

In Exercises \(103-106,\) use the change-of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utility to graph the ratio. $$f(x)=\log _{1 / 2} x$$

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$6 \log _{3}(0.5 x)=11$$

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$5^{x}=212$$

Function \(\quad\) Value $$\text { 58. } f(x)=3 \ln x \quad x=0.74$$

Evaluate \(g(x)=\ln x\) at the indicated value of \(x\) without using a calculator. $$x=e^{-5 / 6}$$
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