Tamil Nadu Board Solutions for Chapter: Basic Algebra, Exercise 12: Exercise

Let $b>0$ and $b\ne 1.$ Express $y=b^x$ in logarithmic form. Also state the domain and range of the logarithmic function.

Prove $\log \frac{a^2}{bc}+\log \frac{b^2}{ca}+\log \frac{c^2}{ab}=0$.

Prove that $\log 2+16\log \frac{16}{15}+12\log \frac{25}{24}+7\log \frac{81}{80}=1.$

Prove ${\log }_{a^2}a{\log }_{b^2}b{\log }_{c^2}c=\frac{1}{8}.$

Prove $\log a+\log a^2+\log a^3+\cdots +\log a^n=\frac{n(n+1)}{2}\log a.$

If $\frac{\log x}{y-z}=\frac{\log y}{z-x}=\frac{\log z}{x-y},$ then prove that $xyz=1.$

If $x=m\sqrt{m}$ is the solution of  ${\log }_{2}x-3{\log }_{\frac{1}{2}}x=6$, then the value of $m$ is

If ${\log }_{5-x}\left(x^2-6x+65\right)=2$ implies $x=-a$ then find the value of $a.$