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Chapter 11: Problem 27

Evaluate the expressions without using a calculator. Verify your answers. Example. We have \(\log _{2} 32=5\) because \(2^{5}=32\). $$ \log _{20} 0.05 $$

Short Answer

Expert verified
Question: Evaluate the logarithmic expression without using a calculator: $$\log_{20}0.05$$ Answer: $$-1$$

Step by step solution

01

Understand the expression

We are given the logarithmic expression \(\log_{20}0.05\). This means we need to find an exponent (let's call it x) that satisfies the equation \(20^{x}=0.05\).
02

Rewrite the number in the expression as a fraction with the same base

We need to rewrite the number 0.05 as a fraction with a base of 20. To do this, we'll first express 0.05 as a fraction: \(0.05=\frac{1}{20}\).
03

Identify the exponent

Now that we have the number as a fraction with the same base (20), we can identify the exponent: \(\frac{1}{20}=20^{-1}\). In this case, the exponent is -1.
04

Write the final answer

Since we found that \(20^{-1}=0.05\), the answer to the original expression is: $$\log_{20}0.05=-1$$

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Key Concepts

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

Exponentiation
Exponentiation is the process of multiplying a number by itself a specific number of times. This number is called the base, and the number of times it is multiplied is called the exponent. For example, with the expression \(2^5\), 2 is the base, and 5 is the exponent. This means we multiply 2 by itself 5 times:
  • \(2 \times 2 = 4\)
  • \(4 \times 2 = 8\)
  • \(8 \times 2 = 16\)
  • \(16 \times 2 = 32\)
Therefore, \(2^5 = 32\). Exponents can also be negative, representing division instead. For example, \(20^{-1}\) means \(\frac{1}{20}\). This concept is useful in the original exercise to understand how converting a base and writing the logarithmic solution works.
Base Conversion
Base conversion is crucial when working with logarithms and exponentiation. It involves converting a number from one base to a different base. In our exercise, we have the expression \(\log_{20}0.05\) and need to express 0.05 in terms of base 20.

To convert decimal 0.05 to a fraction useful in base 20, we rewrite it as \(\frac{1}{20}\). By doing so, we can then identify the exponent needed to reach this fraction when using the base 20, shown as \(20^{-1}\). Effective base conversion lets us find the exponents straightforwardly without needing a calculator or guessing the correct power.
Logarithmic Expressions
Logarithmic expressions can seem daunting, but they simplify the process of finding exponents in a way that relates to real-world applications. The logarithmic form \(\log_{b}(a) = x\) states that the base \(b\) raised to the power of \(x\) equals \(a\): \(b^x = a\). This means when we see \(\log_{20} 0.05\), we are looking for an exponent \(x\) such that \(20^x = 0.05\).

Using our knowledge of exponentiation and base conversion, we realized 0.05 can be expressed as \(\frac{1}{20}\), or \(20^{-1}\). Hence, \(x = -1\). This connects common concepts like multiplication and division to the exponential growth and decay processes observed in math and sciences.

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Most popular questions from this chapter

The size of two towns \(t\) years after 2000 is given by \(u(t)=1200(1.019)^{t}\) and \(v(t)=1550(1.038)^{t} .\) Solve the equation \(u(t)=v(t) .\) What does the solution tell you about the towns?

A trillion is one million million. What is the logarithm of a trillion?

The amount of contamination remaining in two different wells \(t\) days after a chemical spill, in parts per billion (ppb), is given by \(r(t)=320(0.94)^{t}\) and \(s(t)=\) \(540(0.91)^{t} .\) Solve the equation \(r(t)=s(t) .\) What does the solution tell you about the wells?

Without using a calculator, find two consecutive integers, one lying above and the other lying below the logarithm of the number. $$ 0.012 $$

Write the expressions in the form \(\log _{b} x\) and state the values of \(b\) and \(x\). Verify your answers using a calculator as in Example 6 . $$ \frac{\ln 20}{\ln 7} $$
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