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.