Sure! Let's solve the equation [tex]\(10^x = 1,000,000\)[/tex] to find the value of [tex]\(x\)[/tex].
Step-by-Step Solution:
1. Identify the Form of the Equation:
We are given an equation in the form of [tex]\(10^x = 1,000,000\)[/tex].
2. Recognize the Use of Logarithms:
Since we have a power of 10, we can use logarithms to solve for [tex]\(x\)[/tex]. Specifically, we use the common logarithm (base 10).
3. Apply Logarithm to Both Sides:
Take the logarithm (base 10) of both sides of the equation:
[tex]\[
\log_{10}(10^x) = \log_{10}(1,000,000)
\][/tex]
4. Simplify the Left-Hand Side:
Using the property of logarithms [tex]\(\log_{10}(a^b) = b\log_{10}(a)\)[/tex], simplify the left-hand side:
[tex]\[
x \log_{10}(10) = \log_{10}(1,000,000)
\][/tex]
5. Simplify Further:
Since [tex]\(\log_{10}(10)\)[/tex] is 1, the equation simplifies to:
[tex]\[
x \cdot 1 = \log_{10}(1,000,000)
\][/tex]
6. Evaluate the Logarithm:
Calculate [tex]\(\log_{10}(1,000,000)\)[/tex]. We know that [tex]\(1,000,000\)[/tex] is [tex]\(10^6\)[/tex], so:
[tex]\[
\log_{10}(1,000,000) = \log_{10}(10^6)
\][/tex]
7. Use Another Logarithm Property:
Using the same property as before [tex]\(\log_{10}(10^6) = 6 \cdot \log_{10}(10)\)[/tex] and knowing that [tex]\(\log_{10}(10) = 1\)[/tex], we get:
[tex]\[
\log_{10}(10^6) = 6 \cdot 1 = 6
\][/tex]
8. Conclude the Solution:
Therefore, [tex]\(x\)[/tex] equals 6:
[tex]\[
x = 6
\][/tex]
Thus, the exponent that makes the equation [tex]\(10^x = 1,000,000\)[/tex] true is [tex]\(x = 6\)[/tex].
[tex]\(\boxed{6}\)[/tex]
