Algebra Know-It-ALL
Suppose that x is a positive real number. The common logarithm of x can be expressed in terms of the natural logarithms of x and ...
If you have a high-end calculator that goes to two or three dozen digits such as the sort found in some computers, you can leave ...
What Is an Exponential? Theexponential of a quantity is what you get when you raise a certain positive real number, called the b ...
In the first equation, 2 is the argument (the value on which the function depends), and 100 is the resultant. You would say is a ...
for any real number x, and e(lny)=y for any positive real number y. An example Now let’s see how a common antilog can be used to ...
Suppose you have a calculator that can’t directly raise one number to the power of another, but it does have a common log key. Y ...
Considerπ equal to 3.14159, and consider e equal to 2.71828. Then log 10 πe≈ 2.71828 log 10 3.14159 ≈ 2.71828 × 0.49715 ≈ 1.3513 ...
Compare the above with the result of entering −3 into a scientific calculator, then hitting “Inv,” then hitting “ln,” and finall ...
Ratio in an exponent Here’s a more complicated property of exponentials. Let x and y be real numbers, with the restriction that ...
If we let the calculator keep all its extra digits during the calculation process, we don’t get the rounding error when we go ba ...
and the right side becomes (10/e)^4 ≈ (10 / 2.71828)^4 ≈ 3.67880^4 ≈ 183.16 Now let’s invert this ratio. Suppose x is a real num ...
Taking the natural logarithm of each side, we get ln (ex)= ln 1,000,000 x= ln 1,000,000 This simplifies to a matter of finding a ...
Let x= 2.3713018568 and y= 0.902780337. Find xy to three decimal places using common logarithms. Approximate the product xy fro ...
498 CHAPTER 30 Review Questions and Answers Part Three This is not a test! It’s a review of important general concepts you learn ...
The power of product rule allows us to rewrite the left side of this equation, so it becomes [(−1)j]^2 =− 1 But (−1)j is the sam ...
500 Review Questions and Answers The commutative law allows us to rewrite this as ja+jb=j(b+a) and the distributive law, applied ...
Answer 21-9 When we want to multiply one complex number by another, we treat both factors as binomi- als, keeping in mind the fa ...
502 Review Questions and Answers where a and b are coefficients of the variable x, and c is a constant. For the equation to be a ...
Part Three 503 Answer 22-4 This equation is a product of binomials. The roots are the values of x that make either factor equal ...
504 Review Questions and Answers where a and b are coefficients of the variable x,a≠ 0, and c is a constant. What is the general ...
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