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Chapter 10: Problem 41

Use a calculator to approximate each logarithm to four decimal places. $$\log 50$$

Short Answer

Expert verified
1.69897

Step by step solution

01

Understand the Problem

The objective is to approximate the value of \( \log 50 \) to four decimal places using a calculator.
02

Use the Calculator

Enter the number 50 into the calculator. Then, press the 'log' button to find the common logarithm (base 10) of 50.
03

Read the Result

Once you press the 'log' button, the calculator will display the value. Make sure to note this number up to four decimal places.
04

Write the Approximation

Round the result to four decimal places. The value should look something like 1.69897.

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Key Concepts

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

Approximations
Approximations are useful in math when getting an exact value isn't necessary. They're often rounded to a specific number of decimal places for simplicity. In this exercise, we're working with common logarithms, which are logarithms with a base of 10.
When approximating, be sure to round your results correctly. For example, if you're asked to round to four decimal places, like with \(\log 50\), you need to look at the fifth decimal place to determine whether the fourth decimal place rounds up or stays the same.
Common Logarithm
A common logarithm is a logarithm with base 10. This means that \(\log 50\) is asking for the power to which 10 must be raised to get 50. In notation, \(\log_{10} 50\) and \(\log 50\) are the same.
Logarithms can help solve many exponential equations and are widely used in science and engineering. With logarithms, you can transform multiplicative relationships into additive ones, making otherwise complex calculations simpler. For example, in this exercise, we are finding \(\log 50\). This result gives us insight into how 50 relates to the base of 10.
Calculator Usage
Using a calculator to find logarithms is a straightforward process, but attention to detail is crucial. Here鈥檚 how you can correctly approximate \(\log 50\) to four decimal places using a calculator:
  • First, ensure your calculator is in the correct mode. Most often, you'll want the standard mode, not scientific or graphing modes.
  • Next, enter 50, the number you are taking the logarithm of.
  • Press the 'log' button. This key represents the common logarithm function on most calculators.
  • The calculator will display the result. Be sure to record this value accurately.
  • Finally, round the displayed result to four decimal places. For example, \(\log 50\) should be approximately 1.69897, which rounds to 1.6990.

These steps ensure you are using your calculator effectively to find accurate approximations for logarithmic values.

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Most popular questions from this chapter

Use a calculator to approximate each logarithm to four decimal places. $$\log _{5} 26$$

Determine whether common logarithms or natural logarithms would be a better choice to use for solving each equation. Do not actually solve. $$ 10^{3 x+1}=13 $$

Graph each logarithmic function. $$f(x)=\log _{1 / 5} x$$

Choose the correct response. Which statement is false? A. The domain of the function \(f(x)=\left(\frac{1}{4}\right)^{x}\) is \((-\infty, \infty)\). B. The graph of the function \(f(x)=\left(\frac{1}{4}\right)^{x}\) has one \(x\) -intercept. C. The range of the function \(f(x)=\left(\frac{1}{4}\right)^{x}\) is \((0, \infty)\) D. The point (-2,16) lies on the graph of \(f(x)=\left(\frac{1}{4}\right)^{x}\).

Inverse functions can be used to send and receive coded information. A simple example might use the function \(f(x)=2 x+5 .\) (Note that it is one-to-one.) Suppose that each letter of the alphabet is assigned a numerical value according to its position, as follows. $$\begin{array}{llllllllll}\mathbf{A} & 1 & \mathbf{G} & 7 & \mathbf{L} & 12 & \mathbf{Q} & 17 & \mathbf{V} & 22 \\\\\mathbf{B} & 2 & \mathbf{H} & 8 & \mathbf{M} & 13 & \mathbf{R} & 18 & \mathbf{W} & 23 \\\\\mathbf{C} & 3 & \mathbf{I} & 9 & \mathbf{N} & 14 & \mathbf{S} & 19 & \mathbf{X} & 24 \\\\\mathbf{D} & 4 & \mathbf{J} & 10 & \mathbf{O} & 15 & \mathbf{T} & 20 & \mathbf{Y} & 25 \\\\\mathbf{E} & 5 & \mathbf{K} & 11 & \mathbf{P} & 16 & \mathbf{U} & 21 & \mathbf{Z} & 26 \\\\\mathbf{F} & 6 & & & & & & & &\end{array}$$ Using the function, the word ALGEBRA would be encoded as $$\begin{array}{lllllll}7 & 29 & 19 & 15 & 9 & 41 & 7\end{array}$$ because \(f(\mathrm{~A})=f(1)=2(1)+5=7, \quad f(\mathrm{~L})=f(12)=2(12)+5=29, \quad\) and so on The message would then be decoded using the inverse of \(f,\) which is \(f^{-1}(x)=\frac{x-5}{2}\). $$f^{-1}(7)=\frac{7-5}{2}=1=\mathrm{A}, \quad f^{-1}(29)=\frac{29-5}{2}=12=\mathrm{L}, \quad \text { and so on }$$ Use \(f(x)=x^{3}+4\) to encode your name, using the above letter/number assignment
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