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Chapter 13: Problem 25

Use a calculator to verify the given values. $$\log 14+\log 0.5=\log 7$$

Short Answer

Expert verified
The given equation is true. \( \log(14) + \log(0.5) = \log(7) \) simplifies correctly using logarithm properties.

Step by step solution

01

Understanding the Problem

The problem asks to verify if the given equation \( \log(14) + \log(0.5) = \log(7) \) is true by using a calculator. This involves checking whether the left-hand side equals the right-hand side after evaluating both expressions.
02

Applying Logarithm Properties

Recall the logarithmic property: \( \log(a) + \log(b) = \log(ab) \). Apply this property to the expression \( \log(14) + \log(0.5) \), resulting in \( \log(14 \times 0.5) \). Calculate this product to simplify the expression.
03

Calculating Inside the Logarithm

Multiply 14 by 0.5. This gives \( 14 \times 0.5 = 7 \). Hence, \( \log(14 \times 0.5) = \log(7) \). After simplification, the expression becomes \( \log(7) = \log(7) \).
04

Verification with a Calculator

Using a calculator, input and calculate the values for \( \log(14) + \log(0.5) \) and separately calculate \( \log(7) \). Verify that both calculations return the same numerical value.

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Key Concepts

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

Understanding Logarithmic Properties
Logarithmic properties are mathematical rules that help us simplify expressions involving logarithms. One widely used property is the product rule for logarithms:
  • Product rule: \( \log(a) + \log(b) = \log(ab) \)
This rule allows us to combine two logarithmic terms into one by multiplying their arguments. In the original exercise, the expression \( \log(14) + \log(0.5) \) is simplified using this property.
Applying the product rule, we multiply the numbers inside the logs, \( 14 \times 0.5 \), which simplifies to \( 7 \). Therefore, \( \log(14) + \log(0.5) = \log(7) \), showing that both sides of the initial equation are indeed equal. Understanding and using this property can make solving logarithmic equations much simpler.
Verifying with a Calculator
Calculators are powerful tools that assist in verifying mathematical expressions quickly, especially when dealing with logarithms. To confirm if \( \log(14) + \log(0.5) \) equals \( \log(7) \), a calculator can be used to compute both sides.
Here's how you can use it:
  • First, calculate \( \log(14) \) and then \( \log(0.5) \).
  • Add these two results to find \( \log(14) + \log(0.5) \).
  • Separately, calculate \( \log(7) \).
  • Compare the results to check if they are the same.
A calculator provides precise numerical values, ensuring the verification process is swift and straightforward.
Working with Mathematical Expressions
Mathematical expressions are combinations of numbers, variables, and operators that represent a value. In the context of logarithms, expressions like \( \log(14) + \log(0.5) \) involve operations with logarithms.
Understanding how to manipulate these expressions is crucial for solving equations. Here are a few tips:
  • Simplifying expressions: Use properties like the product rule to reduce complex expressions into simpler forms.
  • Checking validity: Convert the expressions using mathematical properties to ensure transformations are valid.
  • Performing calculations: Use a calculator when necessary, especially for non-simple calculations, to avoid errors.
By mastering these techniques, handling and solving problems involving mathematical expressions, especially those with logarithms, becomes more manageable.

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Most popular questions from this chapter

Perform the indicated operations. Using a calculator, solve the equation \(\log _{10} x=x-2\)

$$\text {Plot the indicated graphs.}$$ One end of a very hot steel bar is sprayed with a stream of cool water. The rate of cooling \(R\) (in \(^{\circ} \mathrm{F} / \mathrm{s}\) ) as a function of the distance \(d\) (in in.) from one end of the bar is then measured, with the results shown in the following table. On log-log paper, plot \(R\) as a function of \(d .\) Such experiments are made to determine the hardness of steel. $$\begin{aligned} &\begin{array}{l|l|l|l|l} d \text { (in.) } & 0.063 & 0.13 & 0.19 & 0.25 \\ \hline R\left(^{\circ} \mathrm{F} / \mathrm{s}\right) & 600 & 190 & 100 & 72 \end{array}\\\ &\begin{array}{l|c|c|c|c|c} d \text { (in.) } & 0.38 & 0.50 & 0.75 & 1.0 & 1.5 \\ \hline R\left(^{\circ} \mathrm{F} / \mathrm{s}\right) & 46 & 29 & 17 & 10 & 6.0 \end{array} \end{aligned}$$

If \(x=\ln 4\) and \(y=\ln 5,\) express \(\ln 80\) in terms of \(x\) and \(y\).

$$\text {Plot the indicated graphs.}$$ The acceleration \(g\) (in \(\mathrm{m} / \mathrm{s}^{2}\) ) produced by the gravitational force of Earth on a spacecraft is given by \(g=3.99 \times 10^{14} / r^{2},\) where \(r\) is the distance from the center of Earth to the spacecraft. On log-log paper, graph \(g\) as a function of \(r\) from \(r=6.37 \times 10^{6} \mathrm{m}\) (Earth's surface) to \(r=3.91 \times 10^{8} \mathrm{m}\) (the distance to the moon).

Use a calculator to solve the given equations. $$2^{2 x}-2^{x}-6=0$$
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