Solve for x log base 4 of 5x+2=5x

Solution:

Step 1:

Construct two separate graphs: one for the logarithmic function $y={\log }_4(5x+2)$ and another for the linear function $y=5x$. The x-coordinates where these graphs intersect are the solutions to the equation. The approximate solutions are $x\approx -0.38625003$ and $x\approx 0.14449075$.

Step 2:

There is no further step provided in the original solution. The problem-solving process ends with the graphical method to find the approximate values of x.

Knowledge Notes:

To solve the equation ${\log }_4(5x+2)=5x$, we can use graphical methods as follows:

1. Understanding Logarithms:

   • A logarithm ${\log }_b(a)$ answers the question: "To what power must we raise the base $b$ to obtain $a$?"
   • In this case, ${\log }_4(5x+2)$ asks for the power to which 4 must be raised to get $5x+2$.

2. Graphing the Functions:

   • Graphing involves plotting points on a coordinate system and drawing a curve that passes through these points.

   • For the logarithmic function $y={\log }_4(5x+2)$, we plot points for various values of x and calculate the corresponding y using the definition of logarithms.

   • For the linear function $y=5x$, we plot points by simply multiplying x by 5 to get y.

3. Finding the Intersection:

   • The graphical solution involves finding the point(s) where the two graphs intersect.

   • The x-coordinate(s) of the intersection point(s) are the solution(s) to the original equation.

4. Approximation:

   • Graphical solutions often provide approximate values, especially when dealing with irrational numbers or when intersection points do not fall exactly on grid lines.

5. Graphical Tools:

   • This process can be done by hand on graph paper or by using a graphing calculator or software that can handle plotting and finding intersections of functions.

6. Limitations:

   • Graphical methods may not always provide exact solutions, and the accuracy depends on the scale of the graph and the precision of the plotting tool.

   • For more precise solutions, algebraic methods or numerical methods like the Newton-Raphson method can be used.