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Chapter 4: Problem 36

Evaluate each expression without using a calculator. $$\log _{11} 11$$

Short Answer

Expert verified
The value of the expression \( \log _{11} 11 \) is 1.

Step by step solution

01

Conversion to Exponential Form

The logarithmic expression can be translated to exponential form using the rule \( \log_a b = c \) translates to \( a^c = b \). Applying this here, we get \( 11^{(\log _{11} 11)} = 11 \)
02

Evaluate the exponent

Now, it should be easy to see what the exponent needs to be to make this equation true. As a definition, any number to the power of one equals the number itself. Therefore, for the equation \( 11^{(\log _{11} 11)} = 11 \) to be true, \( \log _{11} 11 \) must equal 1.

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Most popular questions from this chapter

The logistic growth function $$ P(x)=\frac{90}{1+271 e^{-0.122 x}} $$ models the percentage, \(P(x),\) of Americans who are \(x\) years old with some coronary heart disease. Use the function to solve Exercises \(43-46\) What percentage of 80 -year-olds have some coronary heart disease?

In Exercises \(53-56,\) rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm and then round to three decimal places. $$ y=100(4.6)^{x} $$

Explain how to find the domain of a logarithmic function.

Writing in Mathematics Nigeria has a growth rate of 0.025 or \(2.5 \% .\) Describe what this means.

Evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{2} 16\) b. Evaluate: \(\log _{2} 32-\log _{2} 2\) c. What can you conclude about $$\log _{2} 16, \text { or } \log _{2}\left(\frac{32}{2}\right) ?$$
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