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Chapter 12: Problem 10

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log \left(\frac{x}{1000}\right)$$

Short Answer

Expert verified
The expanded and evaluated expression is \(\log(x) - 3\).

Step by step solution

01

- Apply Logarithmic Properties

Begin by applying the logarithmic properties to expand the given logarithmic expression. According to the properties of log mentioned above, express the logarithmic fraction as a difference. Thus, \(\log \left(\frac{x}{1000}\right)\) can be written as \(\log(x) - \log(1000)\).
02

- Evaluate Logarithmic Expression

After expanding the expression, it's now time to evaluate it without the use of a calculator. \(\log(1000)\) is an expression which can be evaluated manually since \(1000 = 10^3\). Thus, \(\log(1000) = 3\). Hence, our expanded and evaluated expression is \(\log(x) - 3\)

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Most popular questions from this chapter

Explain how to use your calculator to find \(\log _{14} 283\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(\log (x+3)=2,\) then \(e^{2}=x+3\)

Explain why the logarithm of 1 with base \(b\) is \(0 .\)

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$\log _{3}(3 x-2)=2$$

Without using a calculator, find the exact value of \(\log _{4}\left[\log _{3}\left(\log _{2} 8\right)\right]\)
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