Solve the given exponential equation by hand. Your answer will involve either the common logarithm or the natural logarithm. (Use log_10 for the common logarithm, and use ln for the natural logarithm.)
10t = 2

To solve the exponential equation:

\[
10^t = 2
\]

we'll take the logarithm of both sides to make it easier to solve for \( t \). You can use either the common logarithm (\(\log_{10}\)) or the natural logarithm (\(\ln\)). Here, I'll use the common logarithm, \(\log_{10}\), since the base is 10.

### Steps:

1. **Take the common logarithm of both sides**:

   \[
   \log_{10}(10^t) = \log_{10}(2)
   \]

2. **Apply the power rule of logarithms**, which states that \(\log_b(a^c) = c \cdot \log_b(a)\):

   \[
   t \cdot \log_{10}(10) = \log_{10}(2)
   \]

3. **Simplify**: Since \(\log_{10}(10) = 1\), the equation becomes:

   \[
   t = \log_{10}(2)
   \]

4. **Write the final answer**:

   \[
   t = \log_{10}(2)
   \]

This is the exact answer in terms of the common logarithm. If you'd like an approximate decimal value:

\[
t \approx 0.3010
\]

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Would you like further explanation on any part of this solution, or do you have any questions?

### Related Questions:

1. How would the solution differ if the equation were \( 10^t = 3 \)?
2. How can we solve \( e^t = 5 \) using natural logarithms?
3. What steps would we take if the equation were \( 2 \cdot 10^t = 6 \)?
4. How can we solve exponential equations if the base is not 10 or \( e \)?
5. What properties of logarithms are useful for solving exponential equations?

### Tip:
Remember, you can always apply logarithms to both sides of an exponential equation to bring the exponent down and solve for it easily.

Solve the given exponential equation by hand. Your answer will involve either the common logarithm or the natural logarithm. (Use log_10 for the common logarithm, and use ln for the natural logarithm.) 10^t = 2

Learn how to solve the exponential equation 10^t = 2 step-by-step by using the common logarithm (log_10). This solution covers key logarithmic concepts and the power rule of logarithms, making it ideal for high school students in Grades 10-12.

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