To solve the equation [tex]\(\log_2 x = \log_5 3 + 1\)[/tex] by graphing, follow these steps:
1. Understand the Equation: Break down the equation into its basic components.
- The left side is a logarithmic function of base 2.
- The right side includes a logarithmic term of base 5 and a constant.
2. Rewrite the Equation: Separate the equation into two parts:
- [tex]\(y_1 = \log_2 x\)[/tex]
- [tex]\(y_2 = \log_5 3 + 1\)[/tex]
3. Use Approximate Values:
- Find the approximate value of [tex]\(\log_5 3\)[/tex]. Since typical calculators may not have a base 5 logarithm function, use the change of base formula:
[tex]\[
\log_5 3 \approx \frac{\log_{10} 3}{\log_{10} 5}
\][/tex]
Using common logarithms (base 10):
[tex]\[
\log_{10} 3 \approx 0.4771 \quad \text{and} \quad \log_{10} 5 \approx 0.6990
\][/tex]
Thus,
[tex]\[
\log_5 3 \approx \frac{0.4771}{0.6990} \approx 0.683
\][/tex]
4. Calculate Right Side:
- Add 1 to [tex]\(\log_5 3\)[/tex]:
[tex]\[
\log_5 3 + 1 \approx 0.683 + 1 \approx 1.683
\][/tex]
- Therefore, [tex]\(y_2\)[/tex] is a constant value approximately equal to 1.683.
5. Graph the Functions:
- [tex]\(y_1 = \log_2 x\)[/tex] is a logarithmic function that increases slowly as [tex]\(x\)[/tex] increases.
- [tex]\(y_2 = 1.683\)[/tex] is a horizontal line.
6. Find the Intersection:
- Plot [tex]\(y_1 = \log_2 x\)[/tex] on a coordinate system.
- Draw the horizontal line [tex]\(y_2 = 1.683\)[/tex].
- The [tex]\(x\)[/tex]-coordinate of the intersection point of these graphs is the solution to the equation.
7. Approximate the Solution:
- By observing or calculating the intersection of these functions:
The intersection point occurs at [tex]\(x \approx 3.2\)[/tex].
Thus, the solution to the equation [tex]\(\log_2 x = \log_5 3 + 1\)[/tex], rounded to the nearest tenth, is:
[tex]\[
x \approx 3.2
\][/tex]
