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Chapter 5: Problem 36

Express as a single logarithm and, if possible, simplify. $$\log 0.01+\log 1000$$

Short Answer

Expert verified
The simplified single logarithm is \(\log(10) = 1\).

Step by step solution

01

Use the product rule for logarithms

The product rule states that \(\log a + \log b = \log (a \cdot b)\). Apply this rule to the given expression \(\log 0.01 + \log 1000\).
02

Multiply the arguments of the logarithms

Multiply the arguments inside the logarithms: \(0.01 \cdot 1000 = 10\).
03

Simplify the logarithm

Rewrite the expression as a single logarithm: \(\log(10)\).
04

Evaluate the logarithm

Recognize that \(\log(10)\) is \(1\) because 10 is the base-10 logarithm.

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Key Concepts

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

Product Rule for Logarithms
When simplifying logarithms, one powerful tool is the product rule. The product rule for logarithms states: \( \log(a) + \log(b) = \log(a \cdot b) \). This property allows you to combine two log expressions into one.
Consider the exercise given, \( \log 0.01 + \log 1000 \), where you need to express these sums as a single logarithm. By applying the product rule, you multiply the values inside the logarithms: \( 0.01 \cdot 1000 \).
The product of these numbers is 10, allowing you to rewrite the expression as \( \log(10) \). This simplification significantly reduces the complexity of the expression and sets up the next steps to further evaluate the logarithm.
Logarithmic Properties
Understanding key logarithmic properties is crucial in simplifying and solving logarithm problems. Here are some essential properties to keep in mind:
  • Product Rule: \( \log(a) + \log(b) = \log(a \cdot b) \).
  • Quotient Rule: \( \log(a) - \log(b) = \log(\frac{a}{b}) \).
  • Power Rule: \( \log(a^b) = b \cdot \log(a) \).
These properties allow you to manipulate and combine logarithms efficiently.
For instance, in the given exercise, the product rule is applied to combine the logs. First, you used the product rule, changing \( \log 0.01 + \log 1000 \) to \( \log(0.01 \cdot 1000) \). This step simplifies the expression to \( \log(10) \). Recognizing and using these properties can turn a seemingly complex problem into a straightforward one.
Evaluating Logarithms
The goal is to simplify and find the value of logarithmic expressions. Once you've applied logarithmic properties, the final step involves evaluating the result.
In our exercise, after using the product rule and simplifying, we have \( \log(10) \). Knowing that the base of common logarithms (\( \log \)) is 10, we can quickly evaluate \( \log(10) \). This simply means finding the power to which 10 must be raised to produce 10. Since 10 raised to the power of 1 is 10, we get \( \log(10) = 1 \).
Therefore, the expression \( \log 0.01 + \log 1000 \) simplifies to 1. Understanding how to evaluate logarithms is the final piece that completes the problem-solving process.

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Most popular questions from this chapter

Retail e-commerce holiday season sales (November and December), in billions of dollars, \(x\) years after 2009 is given by the function $$ H(x)=4.798 x+28.1778 $$ (Source: statista.com). a) Determine the total amount of e-commerce holiday sales in 2012 and in 2016 . b) Find \(H^{-1}(x)\) and explain what it represents. (PICTURE CANNOT COPY)

Use a graphing calculator to find the point \((s)\) of intersection of the graphs of each of the following pairs of equations. $$y=4^{x}+4^{-x}, y=8-2 x-x^{2}$$

Express as a sum or a difference of logarithms. $$\log _{a} \sqrt{9-x^{2}}$$

Determine whether each of the following is true or false. Assume that \(a, x, M,\) and \(N\) are positive. $$\log _{a} M-\log _{a} N=\log _{a} \frac{M}{N}$$

Express as a single logarithm and, if possible, simplify. $$\log _{a}\left(a^{10}-b^{10}\right)-\log _{a}(a+b)$$
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