Solve for x log base 4 of x = log base 8 of 4x

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:

1. 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 ${\log }_b(x)$.

2. 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.

3. 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.

4. Algebraic Solution: To find an exact solution, algebraic manipulation of the equation using the properties of logarithms and other algebraic techniques is necessary.

5. Checking Solutions: After finding a potential solution, it is important to substitute it back into the original equation to verify that it is correct.