Without using a calculator or mathematical table evaluate 2log_10 5-3log_10 2+5log_10 2
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.
To evaluate the expression 2log₁₀5 - 3log₁₀2 + 5log₁₀2 without a calculator or mathematical table, we can use some basic logarithmic properties:
1. logₐ(xy) = logₐx + logₐy
2. logₐ(x/y) = logₐx - logₐy
3. n*logₐx = logₐ(x^n)
Let's rewrite the expression using these properties:
2log₁₀5 - 3log₁₀2 + 5log₁₀2
= log₁₀5² - log₁₀2³ + log₁₀2⁵ (using property 3)
= log₁₀(5²) - log₁₀(2³) + log₁₀(2⁵) (applying rules for multiplying and dividing exponents)
= log₁₀25 - log₁₀8 + log₁₀32 (simplifying the exponents)
= log₁₀(25 * 32) - log₁₀8 (using property 1)
= log₁₀(800) - log₁₀8
= log₁₀(800/8) (using property 2)
= log₁₀(100)
= 2
Therefore, the expression 2log₁₀5 - 3log₁₀2 + 5log₁₀2 evaluates to 2.