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Chapter 4: Problem 21

$$\text { Find each value. If applicable, give an approximation to four decimal places.}$$ $$\log \left(\frac{518}{342}\right)$$

Short Answer

Expert verified
0.1801

Step by step solution

01

- Understand the Logarithm

To solve \(\text{log} \left( \frac{518}{342} \right)\), use the logarithm properties. Here, the argument of the log function is a fraction.
02

- Apply the Quotient Rule

Utilize the quotient rule of logarithms which states \(\text{log} \left( \frac{a}{b} \right) = \text{log}(a) - \text{log}(b)\). Therefore, \(\text{log} \left( \frac{518}{342} \right) = \text{log}(518) - \text{log}(342)\).
03

- Evaluate the Logarithms

Use a calculator to find the values of \(\text{log}(518)\) and \(\text{log}(342)\). \(\text{log}(518) \approx 2.7141\) and \(\text{log}(342) \approx 2.5340\).
04

- Subtract the Values

Subtract the logarithm of the denominator from the logarithm of the numerator. So, \(\text{log}(518) - \text{log}(342) \approx 2.7141 - 2.5340 = 0.1801\).

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Key Concepts

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

Quotient Rule
In logarithms, the quotient rule is a very useful property that can help us simplify complex logarithmic expressions. The quotient rule states that the logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator. Mathematically, this can be expressed as: \( \text{log} \frac{a}{b} = \text{log}(a) - \text{log}(b) \). This means that if you have a logarithm of a fraction, you can separate it into two simpler logarithms. Let's apply this to our exercise, where we have \( \text{log} \frac{518}{342} \). Using the quotient rule, we rewrite this as \( \text{log}(518) - \text{log}(342) \). This makes it much easier to work with, especially when you need to evaluate these logarithms using a calculator.
Logarithmic Properties
Understanding the basic properties of logarithms is crucial for solving problems efficiently. The quotient rule is just one of these properties. Here are a few more that you should know:
  • Product Rule: \( \text{log}(a \times b) = \text{log}(a) + \text{log}(b) \). This means the log of a product is the sum of the logs.
  • Power Rule: \( \text{log}(a^b) = b \times \text{log}(a) \). This means an exponent on the argument can be brought out as a multiplier.
  • Change of Base Formula: \( \text{log}_a(b) = \frac{\text{log}(b)}{\text{log}(a)} \). This is useful for converting between different logarithm bases.
These rules help simplify complex logarithmic expressions and are especially handy when dealing with fractions, products, or powers. Keeping them in mind can make working with logarithms much more straightforward.
Calculator Steps
When evaluating logarithms, a calculator is a helpful tool to find precise values. Here are the steps to follow when calculating logarithms using a standard scientific calculator:
  • Step 1: Turn on your calculator and ensure it's set to the correct mode (typically log base 10 for common logarithms).
  • Step 2: Input the number for which you need the logarithm. For \( \text{log}(518) \), type 518.
  • Step 3: Press the 'log' button. The calculator will display the logarithm value. For example, \( \text{log}(518) \approx 2.7141 \).
  • Step 4: Repeat the process for the other number. For \( \text{log}(342) \), type 342 and press 'log'. This should give you \( \text{log}(342) \approx 2.5340 \).
  • Step 5: Subtract the logarithms by doing 2.7141 - 2.5340 on the calculator, yielding approximately 0.1801.
With these steps, you can quickly and accurately find logarithmic values and solve problems like \( \text{log} \frac{518}{342} \).

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Most popular questions from this chapter

For each function as defined that is one-to-one, (a) write an equation for the inverse function in the form \(y=f^{-1}(x),\) (b) graph \(f\) and \(f^{-1}\) on the same axes, and \((c)\) give the domain and the range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$f(x)=-4 x+3$$

Work each problem. Which of the following is equivalent to \(\ln (4 x)-\ln (2 x)\) for \(x>0 ?\) A. \(2 \ln x\) B. \(\ln (2 x)\) \(\mathrm{C}=\frac{\ln (4 x)}{\ln (2 x)}\) D. \(\ln 2\)

For each function as defined that is one-to-one, (a) write an equation for the inverse function in the form \(y=f^{-1}(x),\) (b) graph \(f\) and \(f^{-1}\) on the same axes, and \((c)\) give the domain and the range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$f(x)=-x^{3}-2$$

Use a graphing calculator to graph each function defined as follows, using the given viewing window. Use the graph to decide which functions are one-to-one. If a function is one-to-one, give the equation of its inverse. $$\begin{aligned}&f(x)=6 x^{3}+11 x^{2}-6;\\\&[-3,2] \text { by }[-10,10]\end{aligned}$$

For each function as defined that is one-to-one, (a) write an equation for the inverse function in the form \(y=f^{-1}(x),\) (b) graph \(f\) and \(f^{-1}\) on the same axes, and \((c)\) give the domain and the range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$y=\frac{4}{x}, \quad x \neq 0$$
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