log10^x = 2a and log10^y = b/2 write 10^a in terms of x

Use the same-derivative argument, as was done to prove the Productand Power Rules for logarithms, to prove the Quotient Ruleproperty.

Derivative of given function using property of logarithms.
${\displaystyle f1x2=\ln \sqrt{10x}}$

Find the exact solution of the exponential equation in terms of logarithms.
Use a calculator to find an approximation to the solution rounded to six decimal places.
I keep getting a negative log answer and decimal...i dont

Computing: ${\displaystyle \underset{x\to 0}{lim}\frac{\log (1+x)}{x^2}-\frac{1}{x}}$
Could the following limit be computed without L'Hopital and Taylor? Thanks.

Removing logs from equation
I have a simple question that I need clarification on:
If
$\log (a)=\log (b)+c$
is it true that
$a=b+\exp (c)$
Is this correct or am I missing something really basic that I cant remember from maths class?

Evaluating a limit with variable in the exponent
For ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}{(1-\frac{2}{x})}^{\frac{x}{2}}}$
I have to use the L'Hospital"s rule, right? So I get:
${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{x}{2}\log (1-\frac{2}{x})}$
And what now? I need to take the derivative of the log, is it: ${\displaystyle \frac{1}{1-\frac{2}{x}}}$ but since there is x, I need to use the chain rule multiply by the derivative of ${\displaystyle \frac{x}{2}}$?

Please, write the logarithm as a ratio of common logarithms and natural logarithms.
${\displaystyle \frac{{\log }_1}{7}\left(4\right)}$
a) common logarithms
b) natural logarithms