We are given:

$\log 2 = 0.3010, \quad \log 3 = 0.4771$

We need to evaluate:

$10^{1.6990}$

Since $10^x$ is the inverse of $\log x$, we recognize that:

$1.6990 = \log x$

Thus, we must express 1.6990 in terms of given logarithmic values:

$1.6990 = \log (2^a \times 3^b)$

Step 1: Express 1.6990 in terms of log values

We assume:

$a \log 2 + b \log 3 = 1.6990$

Substituting the given values:

$a(0.3010) + b(0.4771) = 1.6990$

Step 2: Solve for $a$ and $b$

From trial and calculation, setting $a = 3$ and $b = 2$:

$3(0.3010) + 2(0.4771) = 0.903 + 0.9542 = 1.6990$

Thus:

$10^{1.6990} = 2^3 \times 3^2$

Step 3: Compute the value

$2^3 = 8, \quad 3^2 = 9$

$8 \times 9 = 72$

Final Answer:

$10^{1.6990} = \mathbf{72}$

Would you like a detailed breakdown of any step?

Here are 5 related questions for practice:

1. If $\log 5 = 0.6990$, find $10^{2.0000}$.
2. Express $\log 50$ in terms of $\log 2$ and $\log 5$.
3. Solve for $x$ if $\log x = 1.9542$ given the same log values.
4. Compute $10^{2.3010}$ using $\log 2 = 0.3010$.
5. If $\log 7 = 0.8451$, find $10^{2.8451}$.

Tip: Always check if the given logarithm values can be used directly before attempting approximations!