To rewrite the exponential equation [tex]\( y = 10^x \)[/tex] as a logarithmic function, you can follow these steps:
1. Start with the given exponential equation:
[tex]\[
y = 10^x
\][/tex]
2. Take the logarithm base 10 of both sides of the equation to rewrite it in logarithmic form:
[tex]\[
\log_{10}(y) = \log_{10}(10^x)
\][/tex]
3. Use the logarithmic property that [tex]\(\log_b(a^c) = c \cdot \log_b(a)\)[/tex]. Applying this property:
[tex]\[
\log_{10}(10^x) = x \cdot \log_{10}(10)
\][/tex]
4. Since [tex]\(\log_{10}(10)\)[/tex] equals 1 (because 10 raised to the power of 1 is 10):
[tex]\[
x \cdot \log_{10}(10) = x \cdot 1 = x
\][/tex]
5. Therefore, the equation simplifies to:
[tex]\[
\log_{10}(y) = x
\][/tex]
So, the correct way to write [tex]\( y = 10^x \)[/tex] as a logarithmic function is:
[tex]\[
\log_{10}(y) = x
\][/tex]
