Log Explanations for Homework Statement - Inflation & Property Value

[QuarkCharmer]

Homework Statement

During a 10 year period of constant inflation, the value of a $200,000 property will increase according to the equation v = 200,000e^(.06t).
a.) What will be the value of the property in 4 years?
b.) Use a table/graph to estimate when this property will double in value.

Homework Equations

logb b = 1, logb 1 = 0, logb b^x = x, b^(log x) = x (b != 1, and b > 0)
Product, Quotient, Power properties.

The Attempt at a Solution

a.) This one is simple enough, plug and chug etc.
v = 200,000e^(.06(4))
v = 254249.83 (to 2 decimals)

b.) This one seems easy enough as well.
(2)200,000 = 200,000e^(.06t)
2 = e^(.06t)
ln 2 = ln e^(.06t)
ln 2 = .06 t (because ln e^x = x)
(ln 2)/(.06) = t
t = 12 (to the nearest year)

I have no problem working through these problems, but other than applying the identities that are provided, I have virtually no idea what the log's are doing. Is there some place I can find a better explanation of what is going on. Our book is horrible, and our professor more so. (I do get what eulers no. is, and that ln is just log base e, etc)

Thank you.

 

[symbolipoint]

The nine or eleven properties of Real Numbers which you learn in Elementary Algebra are not fully sufficient to deal with exponential formulas or functions. When we have an the single variable in the exponent, we can take advantage of inverse functions. The natural logarithm function is the inverse of the exponential function.

You might be better able to understand through self-study from another Intermediate or College Algebra textbook.

 

[QuarkCharmer]

I understand what it is.

I am more looking for a proof that (for instance) log(MN) = logM + LogN
From my understanding, log(base b)X = Y is saying "b to the Xth power is Y. I am having some difficulty with problems like this (which I will attempt to work out)

$$log_{6}x + log_{6}10 = 1$$

//use the product property to get a single log

$$log_{6}(10x) = 1$$

//I don't understand exactly what here that makes log(base 6)10x = 1 = 10x = 6?

$$10x = 6$$

$$x = \frac{3}{5}$$

 

[eumyang]

I am having some difficulty with problems like this (which I will attempt to work out)

$$log_{6}x + log_{6}10 = 1$$

//use the product property to get a single log

$$log_{6}(10x) = 1$$

//I don't understand exactly what here that makes log(base 6)10x = 1 = 10x = 6?

You're missing this step:
$$6^{log_{6}(10x)} = 6^1$$

$$10x = 6$$

$$x = \frac{3}{5}$$

 

[eumyang]

I understand what it is.

I am more looking for a proof that (for instance) log(MN) = logM + LogN
From my understanding, log(base b)X = Y is saying "b to the Xth power is Y.

I'll give it a go for the product rule you mention. (Since this proof is not a homework question, I figure it's okay to show the proof here.)
$$\log_b (MN) = \log_b M + \log_b N$$

Let $$x = \log_b M$$ and $$y = \log_b N$$.

Converting these to exponential form,
$$b^x = M$$ and $$b^x = N$$.

So,
$$\begin{aligned} MN &= b^x \cdot b^y \\ MN &= b^{x + y} \\ \log_b (MN) &= x + y \\ \log_b (MN) &= \log_b M + \log_b N \end{aligned}$$

 

[QuarkCharmer]

Thanks, I actually found the explanation of all 3 properties shortly after I posted the thread. I am still concerned with
log(base 6)10x = 1 = 10x = 6?

$$log_{6}10x = 1$$
being equal to
$$10x = 6$$

I don't understand how this step works exactly?

 

[cepheid]

Thanks, I actually found the explanation of all 3 properties shortly after I posted the thread. I am still concerned with
log(base 6)10x = 1 = 10x = 6?

$$log_{6}10x = 1$$
being equal to
$$10x = 6$$

I don't understand how this step works exactly?

If you raise 6 to the power of both sides of the equation, you get:

6^log₆(10x) = 6¹

10x = 6

 

[Outlined]

Ignore log's, use ln's instead. Much easier to understand and (most important) work with. it.

 

[spyrustheviru]

by definition of log(fron now on when I write log I mean log(base6)), logx=y is equal to saying 6^y=x.
to your problem then, you have log(10x)=1, which by definition of log means 6^1=10x or x=3/5.
better now?

 

[Mark44]

Thanks, I actually found the explanation of all 3 properties shortly after I posted the thread. I am still concerned with
log(base 6)10x = 1 = 10x = 6?

Don't write things like the above. Among other things, this is saying that 1 = 6, which is clearly not true.

$$log_{6}10x = 1$$
being equal to
$$10x = 6$$

I don't understand how this step works exactly?

Another way to think about this is that the logarithm of a number is the exponent on the base that produces that number. To restate this in the context of your problem, the logarithm of 10x (shown as 1) is the exponent on 6 that produces 10x.

IOW, 10x = 6¹, from which it follows that x = 3/5.