What does $\log_{10}(1000) = X$ mean?
We start with the expression:
\[\begin{align*} \log_{10}(1000) = X \end{align*}\]
This equation means that we are looking for a value of \(X\) such that:
\[\begin{align*} 10^X = 1000 \end{align*}\]
In other words, \(\log_{10}(1000)\) represents the power to which we must raise 10 to obtain 1000. Our goal is to find this value of \(X\).
Step 1: Rewriting 1000 in Exponential Form
First, observe that $1000$ can be rewritten as a power of 10:
\[\begin{align*} 1000 = 10^3 \end{align*}\]
Thus, we can rewrite our original equation as:
\[\begin{align*} \log_{10}(1000) = \log_{10}(10^3) \end{align*}\]
In general, 1 followed by some number of 0s is always a power of 10. For example,
\[\begin{align*} 1000000 = 10^6 \end{align*}\]
Step 2: Applying the Logarithmic Rule to Solve for \(X\)
We can now use the logarithmic rule that states:
\[\begin{align*} \log_b(b^k) = k \end{align*}\]
Since \(\log_{10}(10^3) = 3\), it follows that:
\[\begin{align*} X = \log_{10}(1000) = 3 \end{align*}\]
So, we find that \( X = \boxed{3} \).
Conclusion
The answer to \(\log_{10}(1000) = X\) is \( X = 3 \). This means that 10 must be raised to the power of 3 to produce 1000. In general, the rule:
\[\begin{align*} \log_b(b^k) = k \end{align*}\]
Can help us to evaluate logarithms when the base and the argument are in the same base-exponent form. This rule is explained in further depth in this article linked here: (link to log(b to the k) article)
Extra Example: Evaluating $\log_{10}(10000)$
To evaluate
\[\begin{align*} \log_{10}(10000) \end{align*}\]
we can use the rule that we’ve been using:
\[\begin{align*} \log_b(b^k) = k \end{align*}\]
Since $10000$ is equal to $10^4$, we can rewrite our expression like so:
\[\begin{align*} \log_{10}(10^4) \end{align*}\]
Since $k = 4$ in this scenario, we find:
\[\begin{align*} \log_{10}(10^4) = \boxed4 \end{align*}\]
