Answer :
To solve the equation [tex]\(10^x = 97\)[/tex], we need to find the value of [tex]\(x\)[/tex]. Here's a step-by-step approach:
1. Understand the Equation: The equation [tex]\(10^x = 97\)[/tex] implies that 10 raised to the power of [tex]\(x\)[/tex] equals 97. To find [tex]\(x\)[/tex], we'll use logarithms because they are the inverse operations of exponentials.
2. Apply the Logarithm: Take the logarithm, base 10, of both sides of the equation:
[tex]\[
\log_{10}(10^x) = \log_{10}(97)
\][/tex]
3. Use Logarithmic Identity: Apply the identity [tex]\(\log_{10}(10^x) = x \cdot \log_{10}(10)\)[/tex]. Since [tex]\(\log_{10}(10) = 1\)[/tex], this simplifies to:
[tex]\[
x \cdot 1 = \log_{10}(97)
\][/tex]
Therefore, [tex]\(x = \log_{10}(97)\)[/tex].
4. Calculate the Logarithm: Compute the value of [tex]\(\log_{10}(97)\)[/tex]. The result is approximately:
[tex]\[
x \approx 1.9867717342662448
\][/tex]
Thus, the exact solution is:
[tex]\[
x = \log_{10}(97) \approx 1.9867717342662448
\][/tex]
Since the calculation has already been performed, you can use this value as the solution.
1. Understand the Equation: The equation [tex]\(10^x = 97\)[/tex] implies that 10 raised to the power of [tex]\(x\)[/tex] equals 97. To find [tex]\(x\)[/tex], we'll use logarithms because they are the inverse operations of exponentials.
2. Apply the Logarithm: Take the logarithm, base 10, of both sides of the equation:
[tex]\[
\log_{10}(10^x) = \log_{10}(97)
\][/tex]
3. Use Logarithmic Identity: Apply the identity [tex]\(\log_{10}(10^x) = x \cdot \log_{10}(10)\)[/tex]. Since [tex]\(\log_{10}(10) = 1\)[/tex], this simplifies to:
[tex]\[
x \cdot 1 = \log_{10}(97)
\][/tex]
Therefore, [tex]\(x = \log_{10}(97)\)[/tex].
4. Calculate the Logarithm: Compute the value of [tex]\(\log_{10}(97)\)[/tex]. The result is approximately:
[tex]\[
x \approx 1.9867717342662448
\][/tex]
Thus, the exact solution is:
[tex]\[
x = \log_{10}(97) \approx 1.9867717342662448
\][/tex]
Since the calculation has already been performed, you can use this value as the solution.
