Question

What is x equal to in the equation log(105⋅10x)=12
log
⁡
(
10
5
⋅
10
𝑥
)
=
12
?

Answer 1

To solve the equation \( \log(105 \cdot 10^x) = 12 \), we can use the properties of logarithms.

First, we can separate the components inside the logarithm:

\[
\log(105 \cdot 10^x) = \log(105) + \log(10^x)
\]

Using the property that \( \log(10^x) = x \), we can rewrite the equation as:

\[
\log(105) + x = 12
\]

Next, we solve for \( x \):

\[
x = 12 - \log(105)
\]

Now we need to calculate \( \log(105) \) using a calculator or logarithm table. The approximate value of \( \log(105) \) (in base 10) is about 2.0212.

Substituting this value into our equation gives:

\[
x = 12 - 2.0212 \approx 9.9788
\]

Therefore, the value of \( x \) is approximately:

\[
\boxed{9.9788}
\]