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Chapter 11: Problem 1

Rewrite the equation using exponents instead of logarithms. $$ \log 0.01=-2 $$

Short Answer

Expert verified
Answer: The exponential form of the given logarithmic equation is \(10^{-2} = 0.01\).

Step by step solution

01

Identify the base and logarithm given in the equation

Since no base is indicated in the equation, we know that the base is 10. The logarithm given in the equation is \(\log{0.01}\) and the exponent is \(-2\).
02

Rewrite equation using the relationship between logarithms and exponents

Using the relationship between logarithms and exponents, we will rewrite the given logarithmic equation in the form of an exponential equation. Remembering that \(\log_b{x} = y\) can be rewritten as \(b^y = x\): $$ 10^{-2} = 0.01 $$ Now, the equation has been rewritten using exponents instead of logarithms.

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Key Concepts

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

Logarithmic Equations
Logarithmic equations might seem challenging at first, but they are just another way to express exponents. Think of logs as questions. A logarithm asks: "To what exponent must we raise the base to get a specific number?" In the equation \(\log 0.01 = -2\), the log is asking what exponent gives 0.01 when using base 10.
  • \(\log_b{x} = y\) reads as "b to what power gives x?"
  • For \(\log 0.01\), the base is 10, though it isn't explicitly written.
  • The equation tells us that 10 raised to the power of -2 equals 0.01.
Recognizing these concepts lets you switch to multiplicative thinking, which is often very handy in solving and understanding similar types of mathematical problems.
Exponents
Exponents are a way to express repeated multiplication of a number by itself. They tell you how many times to use the base number in a multiplication.
  • For example, \(2^3\) means 2 multiplied by itself three times: \(2 \times 2 \times 2 = 8\).
  • With \(10^{-2}\), it denotes the multiplicative inverse: \(1/(10^2)\).
  • Therefore, \(10^{-2} = \frac{1}{100} = 0.01\).
When converting logarithms to exponents, especially with base 10, we often deal with powers of ten that reflect standard decimal notation. Understanding exponents helps simplify working with logarithmic equations.
Base 10
The base 10 system is also known as the decimal system, which we use in our everyday number representation. In logarithmic terms, base 10 logs are simply known as "common logs."
  • When seeing \(\log{x}\) without a base, it generally means base 10.
  • This is the most used base in daily calculations because it aligns with our numerical system.
  • Base 10 is especially useful for scientific work as it easily handles large and small numbers via powers of ten.
For example, converting the logarithmic equation \(\log 0.01 = -2\) to an exponential form \(10^{-2} = 0.01\) uses the base 10 property to easily interpret the decimal value. This usefulness reflects why base 10 is deeply ingrained in mathematics and everyday life.

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