Question

What is x equal to in the equation log(10^5 times 10^x) = 12

Answer 1

We start with the equation:

\[
\log(10^5 \times 10^x) = 12
\]

Using the properties of logarithms, we can simplify the left-hand side. The product property of logarithms states that \(\log(a \times b) = \log(a) + \log(b)\). Therefore, we can rewrite the equation as:

\[
\log(10^5) + \log(10^x) = 12
\]

Next, we can apply the power property of logarithms, which states that \(\log(a^b) = b \cdot \log(a)\). Since the base of our logarithm is 10, we know that \(\log(10) = 1\):

\[
5 \cdot \log(10) + x \cdot \log(10) = 12
\]

This simplifies to:

\[
5 + x = 12
\]

Now, we can solve for \(x\):

\[
x = 12 - 5
\]

\[
x = 7
\]

Thus, the solution is:

\[
\boxed{7}
\]