# Algebra-Functions

### Exponential

solution ${\frac {1}{1\cdot 2}}-{\frac {1}{2\cdot 3}}+{\frac {1}{3\cdot 4}}-{\frac {1}{4\cdot 5}}+........\,$

solution Show that ${\frac {1\cdot 2}{1!}}+{\frac {2\cdot 3}{2!}}+{\frac {3\cdot 4}{3!}}+...........=3e\,$

solution Find the sum of the infinite series ${\frac {1}{2}}-{\frac {1}{2\cdot 2^{2}}}+{\frac {1}{3\cdot 2^{3}}}-.........\,$

solution Find the sum of the infinite series ${\frac {1}{1\cdot 3}}+{\frac {1}{2}}[{\frac {1}{3\cdot 5}}]+{\frac {1}{3}}[{\frac {1}{5\cdot 7}}].........\,$

### Logarithmic

solution If a not equal to 1,m,n are positive real numbers, then show that $\log _{a}({\frac {m}{n}})=\log _{a}(m)-\log _{a}(n)\,$

solution If a not equal to 1 and m,n are positive real numbers, then $\log _{a}(mn)=\log _{a}(m)+\log _{a}(n)\,$

solution If a not equal to 1 and m,k are positive real numbers, then $\log _{a}(m^{k})=k\cdot \log _{a}(m)\,$

solution Rule of change of base,if a and b not equal to 1 and m is a positive real number, then $\log _{a}(m)=\log _{b}(m)\cdot \log _{a}(b)\,$

solution Find the value of $\log _{5}(125)\,$

solution Find the value of $\log _{6}(216{\sqrt {6}}),$

solution Simplify $\log _{{0.01}}(0.00001)\,$

solution Express${\frac {3}{2}}\log(x)-{\frac {1}{3}}\log(y)+{\frac {2}{3}}\log(z)-{\frac {1}{5}}log(a)\,$ as a single logarithm.

solution Find the value of $\log _{3}(4)\cdot \log _{4}(5)\cdot \log _{5}(6)\cdot \log _{6}(7)\cdot \log _{7}(8)\cdot \log _{8}(9)\,$

solution Find the value of $\log _{3}({\sqrt {243}})\,$

solution Find the value of $\log _{{\sqrt {7}}}(343)\,$

solution Find the value of x if $\log _{x}(4)={\frac {-1}{3}}\,$

solution Find the value of x if $\log _{4}(x^{2}+x)-\log _{4}(x+1)=2\,$

solution Find the value of x if $\log _{2}(\log _{2}(x))=1\,$

solution Find the value of x if $\log _{{\sqrt {2}}}(\log _{2}(\log _{2}(x-15))=0\,$

solution Find the value of x if $\log _{2}(\log _{3}(\log _{4}(x))=1\,$

solution Find the value of x if $\log _{5}(x)+\log _{x}(5)={\frac {5}{2}}\,$

solution Find the value of x if $\log _{e}(\log _{e}(\log _{e}(x)))=0\,$

solution Solve ${\frac {1}{2}}\log _{{10}}(11+4{\sqrt {7}})=\log _{{10}}(2+x)\,$

solution If $\log _{x}(k)=a\,$Then find $\log _{{\frac {1}{x}}}(k)\,$ and $\log _{{\frac {1}{x}}}({\frac {1}{k}})\,$

solution Simplify $\log _{{3{\sqrt {2}}}}(5832)\,$

solution If $\log(a+c)+\log(a-2b+c)=2\log(a-c)\,$ then show that a,b,c are in Harmonic Progression.

solution If ${\frac {\log _{2}(x)}{4}}={\frac {\log _{2}(y)}{6}}={\frac {\log _{2}(z)}{3P}},x^{3}y^{2}z=1\,$ then find the value of P.

solution Find the least positive value of x such that$\log _{{\cos x}}\sin x+\log _{{\sin x}}cosx=2\,$

solution If$a=1+\log _{x}(yz),b=1+\log _{y}(zx),c=1+\log _{z}(xy)\,$ then show that ab+bc+ca=abc.

solution If$a^{2}+b^{2}=7ab\,$ then show that $2\log(a+b)=2\log 3+\log a+\log b\,$

solution Find the value of x when $x^{{\log _{3}(x^{2})+(\log _{3}(x))^{{2}}-10}}={\frac {1}{x^{2}}}\,$

solution Solve $a^{{3-x}}\cdot b^{{5x}}=a^{{3x}}\cdot b^{{x+5}}\,$

solution Solve $\log _{2}(9^{{x-1}}+7)=2+\log _{2}(3^{{x-1}}+1)\,$

solution Solve $\log _{{16}}x+\log _{4}x+\log _{2}x=7\,$

solution Solve $(\log _{{10}}(5\log _{{10}}100))^{2}\,$

solution Find x if $\log _{{10}}[98+{\sqrt {x^{2}-12x+36}}]=2\,$

solution Find the value of ${\frac {1}{\log _{{ab}}(abc)}}+{\frac {1}{\log _{{bc}}(abc)}}+{\frac {1}{\log _{{ca}}(abc)}}\,$

solution If ${\frac {\log({\sqrt {x+1}}+1)}{\log({\sqrt[ {3}]{x-40}})}}=3\,$Find the value of x

solution What is the value of$\log _{3}(27\cdot {\sqrt[ {4}]{9}}\cdot {\sqrt[ {3}]{9}})\,$

solution If $(3.7)^{x}=(0.037)^{y}=10000\,$ then find the value of 1/x-1/y

solution Find the value of${\sqrt {10^{{2+{\frac {1}{2}}\log _{{10}}(16)}}}}\,$

solution Find the value of$\log _{{10}}\tan(40)\log _{{10}}\tan(41).......\log _{{10}}\tan(50)\,$

solution Show that $\log _{3}(\log(x^{3}))-\log _{3}(\log x)=1\,$

solution If $\log _{{10}}(x^{2}-x-6)=x+\log _{{10}}(x+2)-4\,$,then find the value of x.

solution If ${\frac {\log x}{2}}={\frac {\log y}{3}}={\frac {\log z}{5}}\,$ then show that $x^{4}=yz\,$

solution Find the value of $\log _{{{\sqrt {a}}}}{\sqrt {a{\sqrt {a{\sqrt {a{\sqrt {a{\sqrt {a}}}}}}}}}}\,$

solution Find the value of $\log _{{{\sqrt {2}}}}{\sqrt {2{\sqrt {2{\sqrt {2{\sqrt {2{\sqrt {2}}}}}}}}}}\,$

solution If $a^{x}=b^{y}=c^{z}=d^{w}\,$ Show that$\log _{a}(bcd)=x\cdot ({\frac {1}{y}}+{\frac {1}{z}}+{\frac {1}{w}})\,$

solution If $a^{2}+b^{2}=6ab\,$ then show that $2\log a+b=\log a+\log b+3\log 2\,$

solution Find the value of ${\frac {\log _{9}11}{\log _{5}13}}-{\frac {\log _{3}11}{\log _{{{\sqrt {5}}}}13}}\,$

solution Solve$\log _{2}x+\log _{4}x+\log _{{16}}x={\frac {21}{4}}\,$

solution If ${\frac {\log a}{b-c}}={\frac {\log b}{c-a}}={\frac {\log c}{a-b}}\,$ prove that $a^{{b+c}}\cdot b^{{c+a}}\cdot c^{{a+b}}=1\,$

solution Find the value of$\log _{a}a+\log _{a}(a^{2})+\log _{a}(a^{3})+........\log _{a}(a^{{2n-1}})\,$

solution If $\log _{{10}}({\frac {1}{2^{x}+x-1}})=x(\log _{{10}}(5)-1)\,$ then find the value of x.