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Exponential Functions

Exponential Functions - Problem Solving


If 3n=5 3^n = 5 and 4m=8 4^m = 8 , what is the value of 9n+m?\large 9^{n+m} ?

Let f(x)=(12)x564, f(x) = \left( \dfrac12 \right)^{x-5} - 64, and suppose that the graphs of y=f(x) y = |f(x)| and y=k y=k meet in the first quadrant.

Find the number of all the positive integers k k that satisfy the above condition.

For all positive integers nn, let αn\alpha_n and βn\beta_n be the roots of the quadratic equation nx2x+n(n+1)=0.nx^2-x+n(n+1)=0. If (1α1+1β1)+(1α2+1β2)++(1α7+1β7)=pq,\left(\frac{1}{\alpha_1}+\frac{1}{\beta_1}\right)+\left(\frac{1}{\alpha_2}+\frac{1}{\beta_2}\right)+ \cdots +\left(\frac{1}{\alpha_{7}}+\frac{1}{\beta_{7}}\right)=\frac{p}{q}, where pp and qq are coprime positive integers, what is p+qp+q?

What is the sum of all possible integer values of xx such that

(x255x+755)(x262x+936)=1? \large (x^2-55x+755)^{(x^2-62x+936 )} =1 ?

Solve for xx :

3=4x+5x+6x 3 = 4^x + 5^x + 6^x


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