Yes, all methods produce the same solution, which is about 1.85. See solution for details.
Practice makes perfect
a
We want to enter the function y=4^x into a calculator. We can do this by pressing the Y= button and typing the equation in the first row.
Having entered the function, by pushing 2ND and GRAPH we get a table of values for whole number inputs of x.
We can see that the solution of 4^x=13 is between 1 and 2. However, we would like to find a solution to this equation with greater accuracy. To do this, we should go to Table Setup. Press 2nd and TBLSET, and then change TblStart to 1.5 and ΔTbl to 0.1.
Now, we can see that the solution of 4^x=13 is between 1.8 and 1.9. However, we would like to find a solution to this equation with greater accuracy. To do this, we can do as before. Press 2nd and TBLSET, and then change TblStart to 1.80 and ΔTbl to 0.01.
Finally, the solution of 4^x=13 is about 1.85.
b
We want to draw graphs of y=4^x and y=13 using a calculator. We can do this by pressing the Y= button and typing the equations in the two first rows.
Having entered the functions, we press the GRAPH button to draw it on a coordinate plane. Also, we can change the viewing window including the scales of x- and y-axis. We can do this by pushing WINDOW.
There is one point of intersection of these lines. To find this point we can use the intersect option, which we get by pushing 2nd and TRACE. Now, we select both graphs and provide the calculator with a guess as to where the intersection might be.
Therefore, the point of intersection is about (1.85, 13).
c
Let's analyze the given equation.
4^x=13
To solve the equation for H, we will use the definition of a logarithm.
b^y= z
⇔
y=log_b z
This definition tells us how to rewrite the logarithm equivalent of y as an exponential equation. The argument z is equal to b raised to the power of y. Let's do it!
4^x= 13
⇔
x=log_4 13
Finally, using a calculator, we get that x=log_413≈ 1.85.
Are all of the methods correct?
Yes. All of the methods produce the same solution, which is around 1.85. Since we are starting with the same equation, all the methods should produce the same result.
