# Rewrite as a sum or difference of multiple logarithms

Only gradually do they develop other shots, learning to chip, draw and fade the ball, building on and modifying their basic swing. In this chapter I explain a suite of techniques which can be used to improve on our vanilla implementation of backpropagation, and so improve the way our networks learn. The discussions are largely independent of one another, and so you may jump ahead if you wish. You can put this solution on YOUR website! There are several properties of logarithms which are useful when you want to manipulate expressions involving them: Used from left to right, this property can be used to separate factors in the argument of a logarithm into separate logarithms. Used from right to left this can be used to combine the sum of two logarithms into a single, equivalent logarithm. Used from left to right, this property can be used to separate the numerator and denominator of a fraction in the argument of a logarithm into separate logarithms.

Used from right to left this can be used to combine the difference of two logarithms into a single, equivalent logarithm. Used from left to right, this property can be used to "move" of the argument of a logarithm out in front of the logarithm as a coefficient.

Used from right to left this can be used to "move" a coefficient of a logarithm into the arguments as the exponent of the logarithm.

This property is used most used from left to right in order to change the base of a logarithm from "a" to "b".

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Since we are interested in separating the x's, y's and z's into separate terms we will be using the first three properties from left to right. Since the argument is a fraction, I'll use property 2 to split the fraction into separate logs: Now I can move the exponent of the argument of the first log out in front using property 3: Now I'll separate the product in the argument of the second log using property 1: Note the parentheses around the new expression.

This is critical since there is a subtraction in front! Next I'll "move" the exponent out the argument of the 3rd log using property 3: And finally I'll subtract the expression in the parentheses:The first part of the TASC Math test consists of 40 multiple choice questions.

Our free TASC Math practice test is a great option for your test prep and review. I am confused about the interpretation of log differences. Here a simple example: \$\$\log(2)-\log(1)\$\$ With my present understanding, I would interpret the result as follows: the number \$2\$ is \$30,10\%\$ greater than \$1,\$ which is obviously false.

Video Example 1: Sum and Difference Properties of Logarithms. Properties of Logs. Sum of Logs Property: log With these properties, we can rewrite expressions involving multiple logs as a single log, or break an expression involving a single log into expressions involving multiple logs.

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms.

(Assume all variables are positive.) 1. 8 xlog 5 §· 2. log 5 x ¨¸ ©¹ 3. lna bc23 4. log 3 b x y 5. 2 3 1 ln, 0 x x x §· ¨¸! ©¹ 6. 4 log 5 xy z 7.

## Math Questions . . . Math Answers . . .

4 24 log x yt 8. 2 ln 2 x y Condense the expression to the logarithm of a single quantity. 9. 2 x 2log 3 xx ln 3ln 1 You can put this solution on YOUR website!

Start with the given expression. Break up the log using the identity Break up the first log using the identity Convert to rational exponent notation. The output of this program is 57, because the first time we print liz her value is 5, and the second time her value is This kind of multiple assignment is the reason I described variables as a container for values.

When you assign a value to a variable, you change the contents of the container, as shown in the figure.

Properties of Logarithmic Functions