Logarithmic and exponential functions: Practice Questions

Using the laws of logarithms, simplify the expression \(2 \log_a 3 + \log_a 4\) to a single logarithm.

Given that \(\log_a 2 = p\) and \(\log_a 5 = q\), express \(\log_2 10\) in terms of p and q.

Find all values of x that satisfy the equation:
\((\log_3 x)^2 - \log_3 (x^2) - 8 = 0\)

Find the value of x that satisfies the equation:
\(\log_2 (x - 3) = 4\)

The solution to the equation \(2^{x+1} = 5^x\) can be written in the form \(x = \frac{\ln 2}{\ln k}\). Find the value of k.

Write the expression \( 3 \log_{10} a + \log_{10} b - 2 \log_{10} c \) as a single logarithm.

Express the following as a single logarithm in its simplest form:
\(\log_a 12 + \log_a 4 - \log_a 6\)

Solve the equation \(\log_2 x + \log_4 x = 6\).

(a) Solve the equation \(2\log_3 x - 〈〉\log_3(x - 2) = 2\).
(b) Solve the equation \(4^y - 〈〉3(2^y) - 4 = 0\).
(c) Use the change of base formula to show that \(\log_a b 〈〉\cdot 〈〉\log_b c 〈〉\cdot 〈〉\log_c a = 1\), where \(a, b, c > 0\) and not equal to 1.

(a) Given that \( y = 3^{2x+1} - 10(3^x) + 3 \), solve the equation \( y = 0 \).

(b) Solve the equation \( \log_2 (x + 3) = 1 + \log_4 (x - 1) \).

(c) A curve has the equation \( f(x) = k \ln(2x - 1) + a \). The curve passes through the point \( (1, 5) \) and the gradient of the curve at this point is 4. Find the value of the constants \( k \) and \( a \).