Indices and Logarithms Problems

Given that $\ln a = x$ and $\ln b = y$, what is the expression for $\ln \frac{a^2b}{e}$ in terms of $x$ and $y$?

What constant value $k$ satisfies $3^x = e^{kx}$ for all $x$?

If $r = \frac{q}{p}$, determine the value of $\frac{\log_q r}{\log_q p}$.

What is the simplified form of the expression $\frac{3^{x+1} \cdot 9^x}{(27^x)^{2/3}}$?

Assuming base 10 logarithm, solve for $x$ in the equation $\log(5x + 6) = 2\log(5x - 6)$.

Given all logarithms are to base 9, simplify the following expression: $\frac{(\log 16)(\log 27)}{(\log 9)(\log 64)}$

Solve the following equation for $x$: $2\log_4 x = \log_4 9$.

Solve for $x$: $2e^{2x} - 5e^x = 12$.

Determine the solution for $x$ in the equation: $\log_a(8 - x) - \log_a(2 - x) = \log_a 3$.

Express the following in its simplest radical form: $\sqrt{\frac{2x^2\sqrt{2}y^5}{5\sqrt{x}y^2}}$

Determine the value of $x$ if $\log_6 x - \log_6 3 = 2$.

Evaluate $2^{3\log_2 4}$.

Solve the equation $3^{2x} = 9^{2-x}$, expressing your answer to three significant figures.

Find the root of the equation $e^{2-2x} = 2e^{-x}$, giving your answer exactly, in terms of logarithms.

Simplify: $\frac{x^5y^{-2}}{(x^3y)^2}$

Simplify the expression: $\log_2 3 \cdot \log_3 4 \cdot \log_4 5 \cdot \log_5 6 \cdot \log_6 7 \cdot \log_7 8$

Simplify $\left( \frac{4^{2x-2}y^{5-3}}{x^2} \right)^{\frac{3}{2}}$

If $x^{3/2} - 5x^{3/4} - 36 = 0$, what is the value of $x$?

Given $\log_a (8 - x) - \log_a (2 - x) = \log_a 3$, solve for $x$.

Given that all logarithms are to the base 9, simplify this expression: $\frac{(\log{16})(\log{27})}{(\log{9})(\log{64})}$

Solve for $x$: $2\log_{4}{x}=\log_{4}{9}$

Express in simplest radical form: $\sqrt{\frac{2x^{2}\sqrt{2}y^{5}}{5\sqrt{x}y^{2}}}$

Given that $\log_{6}{(x)}-\log_{6}{(3)}=2$, solve for $x$.

What is the value of $2^{3\log_{2}{4}}$