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Chapter 15: Problem 27

What is the negative logarithm, \(-\log\), of \(10^{-7}\) ? (Hint: The minus sign in -log means you must put a minus sign in front of the number you get for the logarithm.)

Short Answer

Expert verified
The short answer is: \(-\log_{10}(10^{-7}) = 7\).

Step by step solution

01

Define the negative logarithm formula

Since we are dealing with the negative common logarithm, the formula is: \(-\log_{10}(x) = -\log_{10}(10^{-7})\)
02

Apply the logarithm rule

When the base of the logarithm and the number inside the logarithm are the same, the logarithm of that number is equal to one. Therefore, we can write the logarithm equation as: \(\log_{10}(10^y) = y\)
03

Find the logarithm of the given number

Since \(x = 10^{-7}\), we plug this value into the formula and solve for y: \(\log_{10}(10^{-7}) = -7\)
04

Find the negative logarithm

Now that we know the logarithm is -7, we can find the negative logarithm by applying the hint provided in the question: \(-\log_{10}(10^{-7}) = -(-7) = 7\) The negative logarithm of \(10^{-7}\) is 7.

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Most popular questions from this chapter

Without using a calculator, what is the base-10 logarithm of \(10^{13}\) ?

Use the Bronsted-Lowry definition to explain why \(\mathrm{NH}_{3}\) is a base in water.

How many moles of nitric acid do you need to prepare \(200 \mathrm{~mL}\) of an aqueous solution that has a pH of \(2.0\) ?

What is the \(\mathrm{pH}\) of these aqueous solutions? (a) \(1.0 \mathrm{M} \mathrm{HCl}\), (b) \(0.1 \mathrm{M} \mathrm{HCl}\), (c) \(0.001 \mathrm{M} \mathrm{HCl}\), (d) \(1.0 \times 10^{-5} \mathrm{M} \mathrm{HCl}\), (e) \(1.10 \times 10^{-7} \mathrm{M} \mathrm{HCl}\)

The equilibrium constant for the reaction \(\mathrm{NH}_{4}^{+}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \rightleftarrows \mathrm{NH}_{3}(a q)+\mathrm{H}_{3} \mathrm{O}^{+}(a q)\) is \(5.6 \times 10^{-10}\) (a) Is a solution of ammonium ion very acidic or only slightly acidic? (b) Is water acting as an acid or a base according to the Bronsted-I owry definition? Explain.
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