In order to eliminate the natural log, which has a base of $\displaystyle e$, we will need to raise both side as powers of $\displaystyle e$.

$\displaystyle e^{ln(2x-e)} = e^3$

The equation can be simplified to:

$\displaystyle 2x-e = e^3$

Add $\displaystyle e$ on both sides.

$\displaystyle 2x-e +e= e^3+e$

$\displaystyle 2x= e^3+e$

Divide by two on both sides.

$\displaystyle \frac{2x}{2}= \frac{e^3+e}{2}$

The answer is:  $\displaystyle \frac{e^3+e}{2}$

$\displaystyle 1,000^{\frac{1}{1,000}}$ is the correct choice.

For all $\displaystyle a,N$ for which the expressions are defined, 

$\displaystyle a ^{\frac{1}{N}} = \sqrt[N]{a}$.

Setting $\displaystyle a=1,000, N = 1,000$, this equation becomes

$\displaystyle 1,000 ^{\frac{1}{1,000}} = \sqrt[1,000]{1,000}$ - that is, the one thousandth root of 1,000. This is not equal to 1, since if it were, it would hold that $\displaystyle 1^{1,000} = 1,000$ - which is not true. 

Of the other four expressions:

$\displaystyle \log 10$, the common, or base ten, logarithm of 10, can be rewritten as $\displaystyle \log_{10}10$, and $\displaystyle \ln e$, the natural, or base $\displaystyle e$, logarithm of $\displaystyle e$, can be rewritten as $\displaystyle \log_{e}e$. A property of logarithms states that for all $\displaystyle N > 0, N \ne 1$, $\displaystyle \log_{N}N = 1$. Therefore, $\displaystyle \log 10 = \log_{10}10 = 1$ and  $\displaystyle \ln e = \log_{e}e = 1$.

$\displaystyle 1,000,000^{0} = 1$, since any nonzero number raised to the power of 0 is equal to 1.

By the Power of a Power Property, 

$\displaystyle i^{4} = i ^{2 \cdot 2} =( i^{2} )^{2}$

$\displaystyle i^{2} = -1$, so

$\displaystyle i^{4} = ( -1)^{2}= 1$

For any equation $\displaystyle \log_{10}(y) = x$, $\displaystyle 10^{x } = y$. Thus, we are trying to determine what power of 10 is 1000. $\displaystyle 1000 = 10^3$, so our answer is 3. 

Take the common logarithm of both sides, and take advantage of the property of the logarithm of a power:

$\displaystyle 7 ^{x} = 1,000$

$\displaystyle 7 ^{x} = 10^{3}$

$\displaystyle \log 7 ^{x} = \log 10^{3}$

$\displaystyle x \log 7 = 3 \log 10$

$\displaystyle x \log 7 = 3 \cdot 1$

$\displaystyle x \log 7 = 3$

$\displaystyle \frac{x \log 7}{\log 7} = \frac{3}{\log 7}$

$\displaystyle x = \frac{3}{\log 7}$

Base-10 logarithms are very easy if the operands are a power of $\displaystyle 10$.  Begin by rewriting the question:

$\displaystyle \log{10000000}$

Becomes...

$\displaystyle \log{10^7}$

because $\displaystyle 10^7=10000000$

Applying logarithm rules, you can factor out the $\displaystyle 7$:

$\displaystyle 7\log{10}$

Now, $\displaystyle \log{10}$ is $\displaystyle 1$.

Therefore, your answer is $\displaystyle 7$.

Base-10 logarithms are very easy if the operands are a power of $\displaystyle 10$.  Begin by rewriting the question:

$\displaystyle \log{100000}$

Becomes...

$\displaystyle \log{10^5}$

because $\displaystyle 10^5=100000$

Applying logarithm rules, you can factor out the $\displaystyle 5$:

$\displaystyle 5\log{10}$

Now, $\displaystyle \log{10}$ is $\displaystyle 1$.

Therefore, your answer is $\displaystyle 5$.

Remember:

$\displaystyle log_ab=c$ is the same as saying $\displaystyle a^c = b$.

So when we ask "What is the value of $\displaystyle log(1,000)$?", all we're asking is "10 raised to which power equals 1,000?" Or, in an expression: 

$\displaystyle 10^? = 1,000$.

From this, it should be easy to see that $\displaystyle log(1000)=3$.

Without a subscript a logarithmic expression is base 10.

The expression $\displaystyle \small \log(1000)=\log _{10}(1000)$ 

The logarithmic expression is asking 10 raised to what power equals 1000 or what is x when

$\displaystyle \small 10^{x}=1000$

We know that $\displaystyle \small 10^{3}=1000$

so $\displaystyle \small \log (1000)=3$

Rewrite the logarithm in division.

$\displaystyle \log_{10}10^{3x+2} = \frac{\log{10}^{3x+2}}{\log 10}$

As a log property, we can pull down the exponent of the power in front as the coefficient.

$\displaystyle \frac{\log{(10)}^{3x+2}}{\log( 10)} = \frac{(3x+2)(\log 10)}{\log (10)}$

Cancel out the $\displaystyle \log(10)$.

The answer is:  $\displaystyle 3x+2$

When the base isn't explicitly defined, the log is base 10. For our problem, the first term

$\displaystyle \log 10000=x$

is asking:

$\displaystyle 10^x=10000$

$\displaystyle x=4$

For the second term,

$\displaystyle \log(10)=y$

is asking:

$\displaystyle 10^y=10$

$\displaystyle y=1$

So, our final answer is $\displaystyle 4+1=5$