5(10^x−6)=7

Prove that:
${\displaystyle \underset{x\to 0}{lim}\frac{\ln \left(\cos x\right)}{\ln (1-\frac{x^2}{2})}=1}$
without LHopitals rule.

Arithmetically showing that ${\displaystyle \frac{\log (x+1)}{\log \left(x\right)}<\frac{x+1}{x}}$
Is there a possibility that this can be shown arithmetically? By arithmetically, I mean not looking at the graph.
${\displaystyle \frac{\log (x+1)}{\log \left(x\right)}<\frac{x+1}{x}}$
Thank You

Could you describe this function as "logarithmic"?
${\displaystyle f\left(x\right)=\frac{1}{\sqrt{x}}}$
As x increases, the value of ${\displaystyle f\left(x\right)}$ decreases, but the decrease tapers off quickly as ${\displaystyle x}$ gets larger, and if you plot the graph of ${\displaystyle f\left(x\right)}$, the shape looks kind of like an upside-down logarithm. Would it be correct to describe this function as declining logarithmically, as ${\displaystyle x}$ increases?

How do I evaluate this logarithm expression?
I think the answer for this is 0.01, but I'm not sure. Could someone explain the steps in solving the following for $(x/y)$:
$10{\log }_{10}(x/y)=-20$
I've tried putting $\frac{-20}{10{\log }_{10}}$ in Wolfram Alpha, but the answer doesn't look like what I was expecting.

The One-to-One Property of natural logarithms states that if ln x = ln y, then ________.

Equation ${\displaystyle \log (x^2+2ax)=\log (4x-4a-13)}$ has only one solution; then exhaustive set of values of ${\displaystyle a}$ is

Are there other cases similar to Herglotz's integral ${\int }_0^1\frac{\ln (1+t^{4+\sqrt{15}})}{1+t}\mathrm{d}t$?
This post of Boris Bukh mentions amazing Gustav Herglotz's integral
${\int }_0^1\frac{\ln (1+t^{4+\sqrt{\phantom{A}15}})}{1+t}\mathrm{d}t=-\frac{{\pi }^2}{12}(\sqrt{15}-2)+\ln 2\cdot \ln (\sqrt{3}+\sqrt{5})+\ln \frac{1+\sqrt{5}}{2}\cdot \ln (2+\sqrt{3}).$
I wonder if there are other irrational real algebraic exponents α such that the integral
${\int }_0^1\frac{\ln (1+t^{\alpha })}{1+t}\mathrm{d}t$
has a closed-form representation? Is there a general formula giving results for such cases?
Are there such algebraic $\alpha$ of degree $>2$?