Solve for x log base 4 of x = log base 8 of 4x
The problem you've been given is a logarithmic equation where you are asked to solve for the variable x. Specifically, the equation features two different log base expressions set equal to each other. One is "log base 4 of x," and the other is "log base 8 of 4x." Your task is to find the value(s) of x that satisfy this equation, taking into account the properties of logarithms and the relationship between the bases provided (4 and 8), which are powers of 2.
$\left(log\right)_{4} \left(\right. x \left.\right) = \left(log\right)_{8} \left(\right. 4 x \left.\right)$
Answer
Solution:
Step 1:
Plot the functions \( y = \log_4(x) \) and \( y = \frac{\log_2(4x)}{\log_2(8)} \) on a coordinate system. The x-coordinate where the graphs intersect represents the solution to the equation. Through graphical analysis, we find that the intersection occurs at \( x \approx 16 \).
Step 2:
To verify the solution algebraically, we can use the properties of logarithms to rewrite the equation and solve for \( x \). Since we have already determined the graphical solution, we can check if \( x = 16 \) satisfies the original equation.
Knowledge Notes:
Logarithmic Functions: A logarithmic function is the inverse of an exponential function. The logarithm base \( b \) of a number \( x \) is the exponent by which \( b \) must be raised to yield \( x \). It is denoted as \( \log_b(x) \).
Properties of Logarithms: There are several properties of logarithms that are useful in solving equations:
- Product Property: \( \log_b(mn) = \log_b(m) + \log_b(n) \)
- Quotient Property: \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \)
- Power Property: \( \log_b(m^n) = n \log_b(m) \)
- Change of Base Formula: \( \log_b(m) = \frac{\log_k(m)}{\log_k(b)} \), where \( k \) is a positive real number.
Graphical Solution: Graphing both sides of an equation and finding the point of intersection is a method to visually estimate the solution to the equation. This method is useful when an algebraic solution is difficult to obtain.
Algebraic Solution: To find an exact solution, algebraic manipulation of the equation using the properties of logarithms and other algebraic techniques is necessary.
Checking Solutions: After finding a potential solution, it is important to substitute it back into the original equation to verify that it is correct.
