Exponential and logarithmic functions : แบบฝึกหัด

Solve the equation \( 8^x = 32 \).

Given that \( \log_x 64 = \frac{3}{2} \), find the value of \( x \).

Find the number of distinct real solutions to the equation \(\log_4(2x - 1) + \log_4(x - 2) = 1\).

Solve the equation \( 5^{2x} = 10 \). Express the answer in terms of common logarithms.

If \( \log 2 = p \) and \( \log 3 = q \), express \( \log 72 \) in terms of \( p \) and \( q \).

Find the value of \( \log_3 1 + \log_3 \sqrt{3} \).

Given that \(\log_{10} 2 = a\) and \(\log_{10} 3 = b\), express \(\log_5 18\) in terms of \(a\) and \(b\).

Solve the equation $$4^x - 5 \cdot 2^x + 4 = 0$$.

(a) Simplify \( \frac{x^5 \cdot x^{-1}}{\sqrt[3]{x}} \) and express your answer in the form \( x^n \).
(b) Given that \( \log_5 k = y \), express \( \log_5 \left( \frac{25}{k^2} \right) \) in terms of \( y \).

Consider the expression $E = \log_4 (16x^2)$, where $x$ is a positive real number.

(a) Express $E$ in terms of $\log_2 x$. (2 marks)

(b) Solve the equation $\log_2 x + \log_x 2 = \frac{5}{2}$. (3 marks)