# Mathematics for Computing

Maths

Let a, x and y be real numbers, u and v be positive numbers, and a > 0 with a ≠ 1. Which of the following is/are correct?

(a) log_a (2) = 1
(b) log_a (u/v) = log_a (u-v)
(c) if a^x = y, then x = log_a (y)
(d) log_a (9) = 2log_a (3)
(e) log_a (uv) = log_a (u) + log_a (v)

### Comments for Mathematics for Computing

 Jun 10, 2011 Logarithms by: Staff The question: As you can see, the image you sent is too small to easily read all the detail. It’s especially important to send a larger image when using a *jpg file since a *.jpg file will blur the image (that’s just the way *.jpg compression works). I have tried to sharpen the image a number of different ways, but am still unable to see all of the detail. As closely as I can tell, this is what you submitted: Let a, x and y be real numbers, u and v be positive numbers, and a > 0 with a ≠ 1. Which of the following is/are correct? (a) log_a (2) = 1 (b) log_a (u/v) = log_a (u-v) (c) if a^x = y, then x = log_a (y) (d) log_a (9) = 2log_a (3) (e) log_a (uv) = log_a (u) + log_a (v) The answer: This might help to put logarithms in perspective: Multiplication is a shortcut for addition: 3 + 3 + 3 + 3 = 4 * 3, four times three Exponents are a shortcut for multiplication: 3 * 3 * 3 * 3 = 3⁴, three to the 4th power (the exponent is 4) Logarithms are a shortcut for exponents: log_3 (81) = log_3 (3⁴) = 4, the exponent is 4. The log of 81 using a base of 3 is 4 because the exponent is 4. 81 = 3⁴. (a) log_a (2) = 1 Correct for a=2 (a base of 2). log_2 (2¹) = 1 (b) log_a (u/v) = log_a (u-v) Incorrect Remember that logarithms stand for exponents. The rules of exponents apply to logarithms. For example, 81/3 = (3⁴)/ (3¹) = (3⁴⁻¹) = 3³ log_3 (3³) = 3, the exponent is 3 Therefore, log_3 (81/3) = log_3 (3⁴/3¹) >>>>>>>>>>>>> = log_3 (3⁴) - log_3 (3¹) = 4 - 1 = 3, the exponent is 3 The correct answer to part (b) is: log_a (u/v) = log_a (u) – log_a (v) (c) if a^x = y, then x = log_a (y) Correct. Start with the original equation: a^x = y take the log_a of each side of the equation log_a (a^x) = log_a (y) x = log_a (y) (d) log_a (9) = 2log_a (3) Correct. log_a (9) = log_a (3*3) log_a (9) = log_a (3) + log_a (3) log_a (9) = 2*log_a (3) (e) log_a (uv) = log_a (u) + log_a (v) Correct. Thanks for writing. Staff www.solving-math-problems.com

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