/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 54 Simplify the expression. $$ ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 54

Simplify the expression. $$ \log 0.0001 $$

Short Answer

Expert verified
\(\log(0.0001) = -4\)

Step by step solution

01

Convert to Scientific Notation

Rewrite the number 0.0001 in scientific notation.0.0001 = 10^{-4}
02

Apply Logarithm Property

Use the property of logarithms: \[ \log(a^b) = b\log(a) \]Substitute a and b accordingly: \[ \log(10^{-4}) = -4 \log(10) \]
03

Evaluate Logarithm

Knowing that \log(10) = 1\, we simplify further: \[\log(10^{-4}) = -4 \times 1 = -4 \]Therefore, \log(0.0001) = -4\.

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Key Concepts

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

Scientific Notation
Scientific notation is a way to express very large or very small numbers in a compact form. The general format is to write a number as a product of a number between 1 and 10 and a power of 10. For example, the number 0.0001 can be written in scientific notation as:\[ 0.0001 = 10^{-4} \]
This conversion is helpful because dealing with powers of 10 simplifies many mathematical operations, like logarithms. When you see a problem involving a very small or large number, consider converting it into scientific notation for easier handling. This is our first step in simplifying logarithmic expressions.
Logarithm Properties
Logarithms have several properties that make them very useful for simplifying expressions and solving mathematical problems. One of the key properties you need to know is:\[ \text{log}(a^b) = b \times \text{log}(a) \] This property states that you can move the exponent in front of the logarithm.
In our example, once we converted 0.0001 to scientific notation, we applied this property as follows:\[ \text{log}(10^{-4}) = -4 \times \text{log}(10) \] Knowing these properties can make complex expressions much simpler and manageable.
Logarithm Evaluation
Evaluating logarithms is often straightforward, especially when dealing with base 10 (common logarithms). A key fact to remember is that:\[ \text{log}(10) = 1 \] This is because 10 raised to the power of 1 is 10. Applying this in our example:\[ -4 \times \text{log}(10) = -4 \times 1 = -4 \]
By knowing fundamental evaluations like this, you can easily simplify logarithms in various expressions.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form while retaining their original value. In the context of logarithms and scientific notation, this process often involves several steps: converting to scientific notation, applying logarithm properties, and evaluating basic logarithms.
Taking everything we've learned:\[ \text{log}(0.0001) = \text{log}(10^{-4}) = -4 \times \text{log}(10) = -4 \times 1 = -4 \] Each step contributes to making the expression easier to understand and work with. The goal of simplifying is to make complex mathematical problems more approachable and solvable.

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Most popular questions from this chapter

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log _{3} y+\log _{3}(y+6)=3\)

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(21,000=63,000 e^{-0.2 t}\)

Use the model \(A=P\left(1+\frac{r}{n}\right)^{n t} .\) The variable A represents the future value of P dollars invested at an interest rate \(r\) compounded \(n\) times per year for \(t\) years. If 4000 is put aside in a money market account with interest reinvested monthly at 2.2%, find the time required for the account to earn 1000. Round to the nearest month.

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(e^{2 x}-9 e^{x}-22=0\)

A table of data is given. a. Graph the points and from visual inspection, select the model that would best fit the data. Choose from $$\begin{array}{ll} y=m x+b \text { (linear) } & y=a b^{x} \text { (exponential) } \\ y=a+b \ln x \text { (logarithmic) } & y=\frac{c}{1+a e^{-b x}} \text { (logistic) } \end{array}$$ b. Use a graphing utility to find a function that fits the data. $$ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 2.3 \\ \hline 4 & 3.6 \\ \hline 8 & 5.7 \\ \hline 12 & 9.1 \\ \hline 16 & 14 \\ \hline 20 & 22 \\ \hline \end{array} $$
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