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Chapter 11: Problem 14

Rewrite the equation using logarithms instead of exponents. $$ 10^{-0.08}=0.832 $$

Short Answer

Expert verified
Answer: The given equation rewritten using logarithms is: $$-0.08 = \log_{10}(0.832)$$

Step by step solution

01

Write the given equation

Consider the given equation: $$ 10^{-0.08} = 0.832 $$
02

Take the logarithm of both sides

Since the base of the exponent is 10, we'll take the base-10 logarithm of both sides: $$ \log_{10}(10^{-0.08})=\log_{10}(0.832) $$
03

Use the power rule of logarithms

The power rule of logarithms states that \(\log_{b}(a^x) = x \log_{b}(a)\). Thus, for our equation, we get: $$ -0.08\log_{10}(10) = \log_{10}(0.832) $$
04

Simplify

Since \(\log_{10}(10) = 1\), we can rewrite the equation as: $$ -0.08 = \log_{10}(0.832) $$
05

Solution

So the rewritten equation using logarithms is: $$ -0.08 = \log_{10}(0.832) $$

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Key Concepts

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

Exponents
Exponents are a way to represent repeated multiplication. Think of them as a shorthand for saying how many times you multiply a base by itself. In the equation \(10^{-0.08} = 0.832\), \(10^{-0.08}\) tells us we are using 10 as the base and raising it to a power of \(-0.08\). In general, for any number "a" raised to the power "n", written as \(a^n\), it means:\
    \
  • "a" is the base
    • \
  • "n" is the exponent
    • \
  • This expression equals a multiplied by itself n times (if n is positive)
    • \
\Negative exponents indicate dividing instead of multiplying. For example, \(a^{-n} = \frac{1}{a^n}\). So, \(10^{-0.08}\) actually means \(\frac{1}{10^{0.08}}\), representing a small fraction of the base.
Power Rule of Logarithms
Logarithms transform multiplication into addition, making them incredibly useful in many calculations. One of the essential properties of logarithms is the power rule. This rule states that for any positive numbers "a" and "b", and any real number "x": \\[ \log_{b}(a^x) = x \log_{b}(a) \] \This means that if "a" is raised to the power of "x", we can "bring down" the exponent to simplify the expression. In our problem, using \(10^{-0.08}\), after applying the logarithm, the expression is simplified to: \-0.08 \( \log_{10}(10) \)This simplification process uses the power rule to make logarithmic equations easier to work with.
Base-10 Logarithm
The base-10 logarithm, also called the common logarithm, is a type of logarithm with base 10. The notation for a base-10 logarithm is \( \log_{10}(x) \) and is commonly written as \( \log(x) \) when the base is understood to be 10. In practical terms, this means asking the question: "To what power must 10 be raised to get 'x'?" \
    \
  • For instance, \( \log_{10}(10) = 1 \) because \(10^1 = 10\).
    • \
  • And \( \log_{10}(100) = 2 \) because \(10^2 = 100\).
    • \
\When you apply the base-10 logarithm to the equation \(10^{-0.08} = 0.832\), you are effectively finding out what power 10 must be raised to in order to achieve 0.832, which in this case is -0.08. This simple property makes the base-10 logarithm a handy tool in scientific calculations.

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Most popular questions from this chapter

The number of working lightbulbs in a large office building after \(t\) months is given by \(g(t)=4000(0.8)^{t}\). Solve the equation \(g(t)=1000 .\) What does your answer tell you about the lightbulbs?

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 . $$ \frac{\ln 20}{\ln 7} $$

Evaluate the expressions without using a calculator. Verify your answers. Example. We have \(\log _{2} 32=5\) because \(2^{5}=32\). $$ \log _{3} 9 $$

Without using a calculator, find two consecutive integers, one lying above and the other lying below the logarithm of the number. $$ 0.99 \cdot 10^{5} $$

Write the expressions in the form \(\log _{b} x\) for the given value of \(b\). State the value of \(x\), and verify your answer using a calculator. $$ \frac{4}{\log _{2} 5}, \quad b=5 $$
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