(1) Logarithmic equation with condition

Solve the equation with the defined condition
\( \log_{5}(x - 2) + \log_{5}(x - 3) = 6 \).
Write the solution as a set of real numbers.

(2) Exponential equation with parameters

Solve the equation
\( 3^{x+4} = 5 \cdot 3^{2x} \).
Express the result in the simplest form.

(3) Inverse function

The function is given
\( f(x) = \log_{3}(x - 2) \).
Determine the inverse function and its domain.

(4) Equation with base change

Solve the equation using the base change formula
\( \log_{6} x = \frac{\log x}{\log 2} + 4 \).
Write the solution in exponential form.

(5) Exponential inequality with shift

Solve the inequality
\( 2^{x - 5} \le 3^{x + 4} \).
Display the solution as an interval.

(6) Combination of logarithms

Simplify the expression
\( \frac{\log_{5}(x^{3}) + \log_{5}(x^{4})}{\log_{5} x} \).
Write the result without logarithms where possible.

(7) Growth model and logarithms

The value of the investment is described by the function
\( A(t) = 4000 \cdot e^{5000 t} \).
Determine the time when the value reaches \( 1000 \).

(8) Intersection of functions

Determine the intersection points of the functions
\( f(x) = 3^x \) i \( g(x) = \log_{3}(x + 2) \).
Find the solution analytically or by estimation.

(9) Complex logarithmic equation

Solve the equation
\( \large \log_{3}(x^2 - 4x + 5) = 2 \).
Take into account the defined conditions.

(10) Graph transformation

The function is given
\( f(x) = 5^x \).
Determine the equation of the function resulting from reflection across the \( y \) axis, then a shift of \( 2 \) to the right and \( 4 \) downwards.

(11) Equation with double substitution

Solve the equation
\( 4^{2x} - ( 3 + 5 ) \cdot 4^x + 3 \cdot 5 = 0 \).
Introduce the substitution \( t = 4^x \), solve the quadratic equation for \( t \), and then return to the variable \( x \).

(12) Logarithmic equation with a rational expression

Solve the equation with a full examination of the defined conditions
\( \log_{3} \left( \frac{x - 6}{x - 2} \right) = \log_{3} ( 4 ) - \log_{3} ( 5 ) \).
Write the solution as a set of real numbers and separately list the excluded values.

(13) Parametric exponential equation

Determine all real values of the parameter \( m \) for which the equation has exactly one real solution
\( 4^x + 3 \cdot 4^{-x} = m \).
Then, for the obtained values of the parameter, determine the corresponding solution of the equation.

(14) System of logarithmic and exponential relations

Solve the system of equations
\( y = 5^x \), \( \log_{5}(y) + x = 4 \), \( y > 0 \).
Write the solution as an ordered pair \( (x,y) \).

(15) Proof-problem task with a function

The function is given
\( f(x) = \log_{6}(x + 3) - \log_{6}(5x - 2) \).
Determine the domain of the function, then the equation \( f(x) = 0 \), and examine for which values of \( x \) the condition \( f(x) > 0 \) holds.
Write the conclusion as a union of intervals.