To determine the value of [tex]\(\log_{10}(125)\)[/tex], we can follow these steps:
1. Given the equation [tex]\(10^{0.456} = 5\)[/tex], we can rewrite it using logarithms. This expression tells us that the logarithm base 10 of 5 is 0.456:
[tex]\[
\log_{10}(5) = 0.456
\][/tex]
2. To find [tex]\(\log_{10}(125)\)[/tex], we can use properties of logarithms. Notice that 125 can be expressed as:
[tex]\[
125 = 5^3
\][/tex]
3. Using the logarithm power rule, which states [tex]\(\log_{10}(a^b) = b \cdot \log_{10}(a)\)[/tex], we can rewrite [tex]\(\log_{10}(125)\)[/tex] as follows:
[tex]\[
\log_{10}(125) = \log_{10}(5^3) = 3 \cdot \log_{10}(5)
\][/tex]
4. Substituting the known value of [tex]\(\log_{10}(5)\)[/tex]:
[tex]\[
\log_{10}(125) = 3 \cdot 0.456
\][/tex]
5. Performing the multiplication:
[tex]\[
3 \cdot 0.456 = 1.368
\][/tex]
Therefore, the value of [tex]\(\log_{10}(125)\)[/tex] is [tex]\(1.368\)[/tex].
