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Answer :
To solve the equation [tex]\(10^x = 97\)[/tex], we need to find the value of [tex]\(x\)[/tex] that makes this equation true.
Here is a step-by-step solution:
1. Understand the equation: We have [tex]\(10^x = 97\)[/tex]. This means we are looking for a power of 10 that equals 97.
2. Use logarithms to find [tex]\(x\)[/tex]: To isolate [tex]\(x\)[/tex], we apply the logarithm to both sides of the equation. Using the common logarithm (base 10), we have:
[tex]\[
\log_{10}(10^x) = \log_{10}(97)
\][/tex]
3. Apply the property of logarithms: One of the properties of logarithms is that [tex]\(\log_{10}(10^a) = a\)[/tex], so:
[tex]\[
x \cdot \log_{10}(10) = \log_{10}(97)
\][/tex]
4. Simplify the equation: Since [tex]\(\log_{10}(10)\)[/tex] is 1, the equation simplifies to:
[tex]\[
x = \log_{10}(97)
\][/tex]
Therefore, the exact solution to the equation [tex]\(10^x = 97\)[/tex] is:
[tex]\[
x = \log_{10}(97)
\][/tex]
From calculations, this logarithmic value is approximately [tex]\(1.9867\)[/tex]. So the value of [tex]\(x\)[/tex] that satisfies the equation is:
[tex]\[
x \approx 1.9867
\][/tex]
Here is a step-by-step solution:
1. Understand the equation: We have [tex]\(10^x = 97\)[/tex]. This means we are looking for a power of 10 that equals 97.
2. Use logarithms to find [tex]\(x\)[/tex]: To isolate [tex]\(x\)[/tex], we apply the logarithm to both sides of the equation. Using the common logarithm (base 10), we have:
[tex]\[
\log_{10}(10^x) = \log_{10}(97)
\][/tex]
3. Apply the property of logarithms: One of the properties of logarithms is that [tex]\(\log_{10}(10^a) = a\)[/tex], so:
[tex]\[
x \cdot \log_{10}(10) = \log_{10}(97)
\][/tex]
4. Simplify the equation: Since [tex]\(\log_{10}(10)\)[/tex] is 1, the equation simplifies to:
[tex]\[
x = \log_{10}(97)
\][/tex]
Therefore, the exact solution to the equation [tex]\(10^x = 97\)[/tex] is:
[tex]\[
x = \log_{10}(97)
\][/tex]
From calculations, this logarithmic value is approximately [tex]\(1.9867\)[/tex]. So the value of [tex]\(x\)[/tex] that satisfies the equation is:
[tex]\[
x \approx 1.9867
\][/tex]
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