Question

11. Not all logarithmic and exponential equations can be solved algebraically. Use technology to solve the equation 3

2x−3
=log
2
​
(2x) to 4 decimal places. Include a sketch

Answer 1

To solve the equation 3^(2x-3) = log_2(2x), plot f(x) = 3^(2x-3) and g(x) = log_2(2x) on a graphing calculator and find the intersection point. The x-coordinate of this point, calculated to four decimal places, is the solution.

Step-by-step explanation:

The equation provided is 3^(2x-3) = log_2(2x) and we need to solve for x using technology to four decimal places. To solve this equation numerically, one could use a graphing calculator or software capable of plotting equations or finding numerical solutions.

First, we can express both sides of the equation as a function of x: f(x) = 3^(2x-3) and g(x) = log_2(2x).

We then plot both functions on a graph and look for the intersection point(s). The x-coordinate(s) of the intersection point(s) will give us the solution to the equation. Since the question asks for a visual representation, one would include a sketch of the graph showing the intersection point, but for this text format, we'll simply describe the process.

To solve this specific equation, you would input these functions into a graphing calculator or software, perhaps setting x to range from a lower bound just above zero (as log_2(0) is undefined) to a suitable upper bound where the functions clearly intersect or diverge.

Upon finding the intersection, we use the calculator's or software's capability to determine the x-coordinate to four decimal places, which is the solution to the given equation.

Answer 2

The solution to the combined logarithmic and exponential equation is x = 0.5443 or x = 1.7744 to 4 decimal places.

Solving a combination of logarithmic and exponential equation.

The combination of logarithmic and exponential equation cannot be solved with algebraic method. Using a technology to solve the equation can be through graphical method.

Given that:

`3^(2x-3)=log_2(2x)`

On the graph, we plotted the graph of
`y=3^(2x-3)` and
`y=log_2(2x)`. The point where their curves intersects is the solution to the equation.

From the graph, the values of x are the roots of the equation: Thus, x = 0.5443 or x = 1.7744 to 4 decimal places.