Question

solve \\(\log_{9}5 = 7 - 3x\\) by graphing.

what equation(s) should be graphed?
\\(y_1 = \frac{\log 5}{\log 9}\\)
\\(y_1 = \frac{\log 9}{\log 5}\\)
\\(y_2 = 7 - 3x\\)
\\(y_2 = -3x\\)

graph the system. use the graph to approximate the solution of the equation \\(\log_{9}5 = 7 - 3x\\).
\\(x = \\)

Explanation:

Apply change of base formula

Using the Change of Base Formula knowledge point

$$ \log_{9}5 = \frac{\log 5}{\log 9} $$

Define the system of equations

Using the Graphical Solution of Equations knowledge point

$$
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$$

Solve for x algebraically

We equate the two functions to find the intersection point:

$$ \frac{\log 5}{\log 9} = 7 - 3x $$

Using a calculator to find the decimal value of the logarithmic constant:

$$ \log_{9}5 \approx 0.7325 $$

Substitute this value back into the equation:

$$ 0.7325 \approx 7 - 3x $$

Rearrange to solve for \(x\):

$$ 3x \approx 7 - 0.7325 = 6.2675 $$

$$ x \approx \frac{6.2675}{3} \approx 2.09 $$

Approximate the solution

Rounding the calculated value of \(x \approx 2.09\) to the nearest tenth or integer as typically required by graphical drop-down menus:

$$ x \approx 2.1 $$

Answer:

Question 1

The equations that should be graphed are:

$$y_1 = \frac{\log 5}{\log 9}$$

$$y_2 = 7 - 3x$$

Question 2

The approximate solution to the equation is:

$$x \approx 2.1$$