Solve for x log base 4 of 5x+2=5x
The question is asking you to find the value of the variable "x" which satisfies the logarithmic equation where the logarithm of 5x+2 to the base 4 is equal to 5x. You're expected to use your understanding of logarithms and algebraic manipulation to isolate x and determine what number it must be in order to make the equation true.
$\left(log\right)_{4} \left(\right. 5 x + 2 \left.\right) = 5 x$
Answer
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:
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 $5x + 2$.
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.
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.
Approximation:
- Graphical solutions often provide approximate values, especially when dealing with irrational numbers or when intersection points do not fall exactly on grid lines.
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.
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.
