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2×log3 27−5×log8 64+10×log10 1000

We can solve the expression step by step, simplifying each logarithmic term:

\[
2 \times \log_3{27} - 5 \times \log_8{64} + 10 \times \log_{10}{1000}
\]

### Step 1: Simplify each logarithm
#### First term: \( \log_3{27} \)
Since \( 27 = 3^3 \), we can use the property \( \log_b{(b^k)} = k \):
\[
\log_3{27} = \log_3{(3^3)} = 3
\]
Now, multiply by 2:
\[
2 \times 3 = 6
\]

#### Second term: \( \log_8{64} \)
We know \( 64 = 8^2 \), so:
\[
\log_8{64} = \log_8{(8^2)} = 2
\]
Now, multiply by -5:
\[
-5 \times 2 = -10
\]

#### Third term: \( \log_{10}{1000} \)
Since \( 1000 = 10^3 \), we have:
\[
\log_{10}{1000} = \log_{10}{(10^3)} = 3
\]
Now, multiply by 10:
\[
10 \times 3 = 30
\]

### Step 2: Combine all terms
Now that we have simplified each part, combine the results:
\[
6 - 10 + 30 = 26
\]

Thus, the value of the expression is:

\[
\boxed{26}
\]

Would you like further details or clarifications?

Here are 5 related questions:
1. How can we generalize the properties of logarithms for different bases?
2. Can logarithmic properties be extended to complex numbers?
3. What are some real-world applications of logarithms in engineering or science?
4. How do logarithmic scales, like the Richter scale, function?
5. How would this calculation change if the logarithms were natural logs?

**Tip:** When simplifying logarithmic expressions, always try to rewrite the arguments as powers of the base to make the calculations easier.

2×log3 27−5×log8 64+10×log10 1000

Learn how to simplify the logarithmic expression 2×log3 27−5×log8 64+10×log10 1000 step by step, using properties of logarithms like log_b(b^k) = k. This problem covers key concepts such as logarithmic multiplication and the change of base theorem. Suitable for high school students in Grades 9-12.

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