- #1
DCHomage
- 16
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Homework Statement
Write expression as a logarithm of a single quantity and then simplify if possible.
(1/4)[log (x²-1)-log (x+1)]+3log x
Homework Equations
The Attempt at a Solution
I got (1/4)log(x-1) + 3log x so far
DCHomage said:Not really. i need to know the steps towards the answer.
gabbagabbahey said:Well, what does your first term become when you use the property in my previous post with [itex]a=\frac{1}{4}[/itex] and [itex]b=(x^2-1)[/itex]?
gabbagabbahey said:There was a typo in my last post, I meant [itex]b=(x-1)[/itex] (without the squared!), which would give you:
[tex]\frac{1}{4} Log(x-1)=Log((x-1)^{1\over{4}})= Log(\sqrt[4]{x-1})[/tex]
What do you get for your second term using the same rule (with a different [itex]a[/itex] and [itex]b[/itex] this time of course)?
gabbagabbahey said:The 3log(x) is your other term...
I'm not going to just show you the steps, it is important for your future that you learn to think through these problems on your own.
Using the rule in my first reply, what can you express 3log(x) as?
gabbagabbahey said:Yes, good.
Now if [itex]\frac{1}{4} Log(x-1)= log(\sqrt[4]{x-1})[/itex] and [itex]3log(x)=log(x^3)[/itex], what is [itex]\frac{1}{4} Log(x-1)+3log(x)[/itex]?
gabbagabbahey said:Do you mean [itex]log(x^3 \sqrt[4]{x-1})[/itex] or [itex]x^3log(\sqrt[4]{x-1})[/itex]?
gabbagabbahey said:Looks good to me.
...If your not sure about your result, try plugging a few random numbers for x into your starting expression, and final expression (with the help of a calculator); you should find that they both give the same value for a given x.
gabbagabbahey said:That's as far as you can simplify it...you have gone from having an expression with 3 terms, to an expression with only 1 term, so you have simplified it a fair bit.
gabbagabbahey said:[itex]log(x^3 \sqrt[4]{x-1})[/itex] looks to me like a more simple expression than [itex]\frac{1}{4}[log (x^2-1)-log (x+1)]+3log(x)[/itex]...would you not agree?
If so, then you've certainly simplified it at least a little. And I don't see any way to simplify it any more than that...do you?
If not, then your done: [itex]log(x^3 \sqrt[4]{x-1})[/itex] is your final answer!
Do you follow?
A single logarithm is a logarithm in which there is only one variable. This means that the logarithm contains only one term, rather than multiple terms being added or subtracted.
Turning to a single logarithm allows for easier evaluation and simplification of logarithmic expressions. It also helps to make solving equations involving logarithms more straightforward.
To turn multiple logarithms into a single logarithm, you can use the properties of logarithms to combine them into one. For example, if there are two logarithms being added, you can use the product property of logarithms to combine them into one logarithm with the product of the two terms inside.
Yes, there are several rules and properties of logarithms that can be used when turning to a single logarithm. These include the product property, quotient property, power property, and the fact that the logarithm of a number to its base is equal to 1.
Sure, for example, if we have the expression log(x) + log(y), we can use the product property to turn it into a single logarithm: log(xy). Similarly, if we have the expression log(x) - log(y), we can use the quotient property to turn it into a single logarithm: log(x/y).