Start typing, then use the up and down arrows to select an option from the list. ## General Chemistry

Learn the toughest concepts covered in Chemistry with step-by-step video tutorials and practice problems by world-class tutors

5. BONUS: Mathematical Operations and Functions

# Power and Root Functions

Logarithmic and natural logarithmic functions of numbers.

Logarithmic Functions

The logarithmic base 10 form represents the exponent that 10 must be raised in order to obtain that specific number.

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so, without using a calculator, determine the answer to the following questions. So here we have long of 1.0 times 10 to the five. So just like up above, the log is getting distributed to the one end to the 10 to the fifth. So we're gonna say here because they're multiplying. What this really means is we have long of one plus log off 10 to the fifth. Remember up above, we said that log of one is equal to zero, and then this is logged based 10. So this cancels out with this, giving us five at the end. So here, log of that number would be five. Now, this is easy to remember in terms of figuring out the log of an answer really quickly. But if this numbers happens to be a number different from one, then it's best toe and put it into your calculator. Because if it's log of any other number, this part won't be equal to 10 and you haven't memorized what those numbers are because you have your calculator. So again, this works best when we have long at 10 times 10 to any power, and then finally this one here we have a log of 10. We want to change this. So we're going to say that if we move this over and change to scientific notation, then we have log off 10 times 10 to the negative four. So it just becomes like the one we just did. The law gets distributed to both. So this is log of one plus log off. 10 to the negative. Four long of one is just zero. This is base 10 which cancels out with this, which gives me negative four. So it's zero plus negative four. So the answer he would be negative for is my final answer. So these are just the basics. When it comes to log functions, they're gonna become important when we're talking about chapters dealing with chemical kinetics also will come in handy when we're talking about determining Ph or P o. H off different solutions. That's when log functions really come out. And we have to do different types of mathematical manipulations to get our answer at the end. But for now, just remember the basics when it comes toe these blogged functions here
Inverse Logarithmic Functions

The inverse or anti-logarithmic function is the opposite of the logarithmic function.

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Natural Logarithmic Functions

The natural logarithmic function ln is the exponent to which e must be raised to determine that number.

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so the natural logarithms sometimes shortened to just Ln so Alan is the natural longer rhythm off a value is the exponents which e must be raised to determine that number. So, for example, Ellen of 1000 equals 6.908 e is the inverse of the natural log. So we do e to 6.908 That would give me 1000. Okay, so here, the inverse of the natural logarithms again represent, symbolized by the variable e can be seen as the opposite of the natural logarithms. And so, Alan of X equals y. So the inverse of my natural law y equals e to the y equals X. All right, so So applying this logic, the example below would go as follows. So we have Ellen of two equals 20.693 So the inverse of my Ln inverse of my natural law, which is e to the 0.693 equals two. Now, when do we have to deal with this natural logarithms? We're going to say that this is very common when we deal with chemical kinetics. So we deal with the chapter dealing with chemical kinetics. You will see Ln being used now, using this logic, we can help to solve this question. Here, here it says, based on your understanding of natural law, go rhythmic functions solve for the missing variable in the following question. So they give us Ellen of acts equals negative. 2.13 times 10 to the negative one times 12. plus Allen of 1 to 5. So we're gonna do here is we're gonna bring down Al Innovex. We're going to multiply these together. Don't forget the negative sign that's there. So we multiply those two together we get negative 26199 Then we're gonna add Allen of 1 to 5 to that. So punch in on your calculator Ln off 1. in your calculated, you might have to hit second function. Look for Ln and then parentheses. 1.25 close parentheses. When you do that, you'll get 0. And then when we add those together we get negative 2.39676 which is still equal toe Elena backs. We need to figure out what X is to do that we have to take the inverse of my natural logarithms. So here, when we take the inverse, that'll get rid of my Ln and then it's gonna be e to this value. So whatever the value is, it gets raised to a power. So when you're calculating, you may see this button e to the X. Usually you have to hit second function on your calculator, then hit that button, then parentheses. Plug in negative. 239676 Close parentheses. If you do it correctly, what you're gonna get is 2396760. That would be our value. Here, here, we're not gonna worry about Sig Figs. Let's just worry about getting the correct answer. So this will be my value for X Again. We use the inverse of the natural law algorithm if we have Ln X equals some value. To find the X, we have to get rid of the l N. And we do that by taking the inverse of both sides taking the inverse gets rid of the L N. And then it raises this number. Tow a power. It becomes e to that number that will help us find our missing variable
Logarithmic Relationships

The similarities between logarithmic and natural logarithmic functions are outlined below.

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