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Logarithm Notes

Facts about Logarithms

|Invented in early 1600’s. Has numerous applications in finance, math and science. Prior to calculators the slide rule, based on |

|logarithms, allowed for quick arithmetic computations. |

|A logarithm is an exponent |

|If bm=x, then m = logb x |

|Example: if 32 = 9, then log3 9 = 2 |

|Used to solve exponential equations |

|Log functions are inverses of exponential functions |

|y = loga x is the inverse of y=ax |

|Example: y = log5 x is the inverse of y = 5x. |

|To illustrate this exchange x & y, we get x = log5 y, so y = 5x |

|Properties |Example |

|logb b = 1 & logb 1 = 0 & logb 0 = und |log7 7 = 1 and log7 1 = 0 and log7 0 = und |

|2. (product rule) |log7 15 = log7 5 + log7 3 |

|logb xy = logb x + logb y | |

|(quotient rule) |log7 [pic] = log7 5 - log7 3 |

|logb [pic] = logb x - logb y | |

|(power rule) |log7 [pic]= log7 [pic]= [pic]log7 5 |

|logb xr = r logb x | |

| |Special case log7 75 = 5 |

|Special case logb bx = x | |

| | |

|Properties used to solve log equations: | |

|a. if bx = by, then x = y | |

|b. if logb x = logb y, then x = y |a. if 3x + 3 = 34, then x + 3 = 4 x = 1 |

|Notes: |b. if log7 x + 3 = log7 4, then x + 3 = 4 x = 1 |

|i. x > 0 and y >0 must be true | |

|ii. converse is also true | |

Natural Logarithms

|Definition |ln x = loge x |

|Value of e |e1 [pic] 2.72 defined as [pic] |

|Function |y = ln x |

|Inverse |y = ex to illustrate switch x & y so x = ln y = loge y and y = ex ⎫ |

|Property |ln (ex) = x to illustrate loge ex = x loge e = x |

|Other Properties |All of the above properties apply |

PROOF OF PROPERTIES

|Property |Proof |Reason for Step |

|logb b = 1 and logb 1 = 0 |b1 = b ⎫ and b0 = 1 ⎫ |Definition of logarithms |

|2. (product rule) |a. Let logb x = m and logb y = n |a. Setup |

|logb xy = logb x + logb y |b. x = bm and y = b n |b. Rewrite in exponent form |

| |c. xy = bm * bn |c. Multiply together |

| |d. xy = b m + n |d. Product rule for exponents |

| |e. logb xy = m + n |e. Rewrite in log form |

| |f logb xy = logb x + logb y ⎫ |f. Substitution |

|(quotient rule) |a. Let logb x = m and logb y = n |a. Given: compact form |

|logb [pic] = logb x - logb y |b. x = bm and y = b n |b. Rewrite in exponent form |

| |c. [pic] = [pic] | |

| |d. [pic]= [pic] |c. Divide |

| |e. logb [pic]= m - n | |

| |f. logb[pic] logb x - logb y ⎫ |d. Quotient rule for exponents |

| | | |

| | |e. Rewrite in log form |

| | | |

| | |f. Substitution |

|(power rule) |a. Let m = logb x so x = bm |a. Setup |

|logb xn = n logb x |b. xn = bmn |b. Raise both sides to the nth power |

| |c. logb x n = mn |c. Rewrite as log |

| |d. logb xn = n logb x |d. Substitute |

|5. Properties used to solve log equations: | | |

| | | |

|a. if bx = by, then x = y | | |

| |a. This follows directly from the properties for | |

| |exponents. | |

|b. if logb x = logb y, then x = y | | |

| |b. i. logb x - logb y = 0 |b. i. Subtract from both sides |

| |ii. logb[pic] | |

| |iii. [pic]= b0 |ii. Quotient rule |

| |iv. [pic]1 so x = y | |

| | | |

| | |iii. Rewrite in exponent form |

| | | |

| | | |

| | |iv. b0 = 1 |

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