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Chapter 10: Problem 60

Write each logarithmic statement in exponential form. $$ \log _{10} 0.001=-3 $$

Short Answer

Expert verified
The exponential form is \( 10^{-3} = 0.001 \).

Step by step solution

01

Identify the Given Logarithmic Statement

Given the logarithmic statement: \ \( \log_{10} 0.001 = -3 \)
02

Understand the Definition of Logarithms

Recall that a logarithmic statement \( \log_{b} a = c \) can be rewritten in its exponential form as \( b^c = a \).
03

Rewrite the Given Logarithmic Equation

According to the form \( \log_{b} a = c \), identify \( b = 10 \), \( a = 0.001 \), and \( c = -3 \). Substitute these values into the exponential form \( b^c = a \).
04

Convert to Exponential Form

Rewrite the statement \( \log_{10} 0.001 = -3 \) in exponential form: \ \( 10^{-3} = 0.001 \).

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Key Concepts

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

logarithmic equations
Logarithmic equations are mathematical expressions involving logarithms. A logarithm answers the question, 'To what power must a base be raised, to yield a specific number?' For example, in the equation \(\text{log}_{10} 0.001 = -3\), we are saying that \(10^{-3}=0.001\). The given logarithmic statement can be converted to an exponential form, making it easier to understand the relationship between the numbers involved.
exponential form
Converting a logarithmic equation to exponential form is an essential skill in algebra. When we see \(\text{log}_{b} a = c\), this translates to \(b^c = a\) in exponential form. For the given problem, \(10^{-3}=0.001\), which shows how \(0.001\) is derived. Recognizing this helps in understanding and solving logarithmic problems more effectively.
properties of logarithms
To understand logarithms deeper, it's crucial to be familiar with their properties:
\- The **Product Property** states that \(\text{log}_{b}(xy) = \text{log}_{b}(x) + \text{log}_{b}(y)\).
\- The **Quotient Property** states that \(\text{log}_{b}(\frac{x}{y}) = \text{log}_{b}(x) - \text{log}_{b}(y)\).
\- The **Power Property** states that \(\text{log}_{b}(x^y) = y \text{log}_{b}(x)\). These properties are valuable for manipulating logarithmic expressions, making them easier to solve or convert to exponential form.
base 10 logarithms
Base 10 logarithms, or common logarithms, use 10 as the base. They are often written as \(\text{log}_{10}(x)\) or simply \(\text{log}(x)\) with the base 10 implied. In our example \(\text{log}_{10} 0.001 = -3\), we find that raising 10 to the power of \(-3\) results in \(0.001\). These logarithms are widely used in science and engineering because they simplify the calculations involving powers of 10.

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Most popular questions from this chapter

A study showed that the number of mice in an old abandoned house was approximated by the function defined by $$ M(t)=6 \log _{4}(2 t+4) $$ where \(t\) is measured in months and \(t=0\) corresponds to January \(2008 .\) Find the number of mice in the house in (a) January 2008 (b) July 2008 (c) July 2010 (d) Graph the function.

Let \(m\) be the number of letters in your first name, and let \(n\) be the number of letters in your last name. (a) In your own words, explain what \(\log _{m} n\) means. (b) Use your calculator to find \(\log _{m} n .\) (c) Raise \(m\) to the power indicated by the number found in part (b). What is your result?

A major scientific periodical published an article in 1990 dealing with the problem of global warming. The article was accompanied by a graph that illustrated two possible scenarios. (a) The warming might be modeled by an exponential function of the form $$y=\left(1.046 \times 10^{-38}\right)\left(1.0444^{x}\right)$$ (b) The warming might be modeled by a linear function of the form $$y=0.009 x-17.67$$ In both cases, \(x\) represents the year, and y represents the increase in degrees Celsius due to the warming. Use these functions to approximate the increase in temperature for each of the following years. $$ 2040 $$

The number of years, \(N(r),\) since two independently evolving languages split off from a common ancestral language is approximated by $$N(r)=-5000 \ln r$$ where \(r\) is the percent of words (in decimal form) from the ancestral language common to both languages now. Find the number of years (to the nearest hundred years) since the split for each percent of common words. (a) \(85 \% \text { (or } 0.85)\) (b) \(35 \% \text { (or } 0.35)\) (c) \(10 \% \text { (or } 0.10)\)

Decide whether each statement is true or false. $$ \log _{2}(8+32)=\log _{2} 8+\log _{2} 32 $$
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