Here are ten challenging problems that involve exponential & logarithmic expressions. The problems are not in any particular order (e.g. difficulty). You can download a document with the problems (and answers) using the link below.

Exponents & Logarithms - Challenge Qs

1.Given that $x={\left({\log }_{8}2\right)}^{\left({\log }_{2}8\right)}$ , find the value of ${\log }_{3}x$ .

2.Given $\frac{4^x}{2^{x+y}}=8$ and $\frac{9^{x+y}}{3^{\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}5y}}=243$ , $x,y\in ℝ$ , find the value of $xy$ .

3.Given ${\log }_{M}N={\log }_{N}M,M\ne N,MN>0,M\ne 1,N\ne 1$ , find the value of $MN$ .

4.Consider that $\frac{\log a}{p}=\frac{\log b}{q}=\frac{\log c}{r}=\log x$ with all the logarithms having the same base and $x\ne 1$ .
If $\frac{b^2}{ac}=x^y$ , then what is the value of $y$ .

5.Find $\underset{x\to +\infty }{\lim }\left[{\log }_3\left(\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}6x-5\right)-{\log }_3\left(\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}2x+1\right)\right]$ .

6.Given $\frac{{64}^{x-1}}{4^{x-1}}={256}^{\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}2x}$ , find the real value of $x$ .

7.Solve exactly for $x$ : $2^{\mathrm{}\mathrm{}\mathrm{}2x}-8\cdot 2^x+12=0$

8.Given $x=1+2^p$ and $y=1+2^{-p}$ , then find $y$ in terms of $x$ .

9.Given $a={\log }_{8}225$ and $b={\log }_{2}15$ , then find $a$ in terms of $b$ .

10.If $3^{\mathrm{}\mathrm{}2x}+9=10\left(3^x\right)$ , then find the value(s) of $x^2+1$ .