Without using a calculator or mathematical table evaluate 2log_10 5-3log_10 2+5log_10 2

Answer 1

To solve the expression without using a calculator or mathematical table, we can work on simplifying it step by step using logarithmic properties:

2log₁₀ 5 - 3log₁₀ 2 + 5log₁₀ 2

Using the logarithmic property logₐ(b^c) = clogₐ(b), we can write:

log₁₀ (5²) - log₁₀ (2³) + log₁₀ (2⁵)

Using the logarithmic property logₐ(b) - logₐ(c) = logₐ(b/c), we can write:

log₁₀ (5²) - log₁₀ ((2³)/(2⁵))

Using the logarithmic property logₐ(b^n) = nlogₐ(b), we can write:

log₁₀ (25) - log₁₀ ((2³)/(32))

Using the logarithmic property logₐ(b) + logₐ(c) = logₐ(b * c), we can write:

log₁₀ (25/(2³ * 32))

Simplifying further:

log₁₀ (25/2⁹)

Then, using the logarithmic property logₐ(b * c) = logₐ(b) + logₐ(c), we can write:

log₁₀ (25) - log₁₀ (2⁹)

Finally, we can write the expression as:

2(log₁₀ (25)) - 9(log₁₀ (2))

Now, it's important to note that log₁₀(25) = 2 and log₁₀(2) is approximately 0.301.

Plugging these values into the expression, we get:

2(2) - 9(0.301)

Which simplifies to:

4 - 2.709

Therefore, the value of the expression without using a calculator or mathematical table is approximately 1.291.