prove that:log25+log8*log16/log64 -log14/5+log28=3
put it in your calculator and see if it works
i donot have calculator now
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log8 * log16/log64
= log 2^3 * log 2^4 / log 2^6
= 3log2 * 4log2 / 6log2
= 2log2
= log 4
log (14/5) = log14 - log 5
and log 25 = log 5^2 = 2log 5
so
log25+log8*log16/log64 -log14/5+log28
= log 25 + log 4 - (log14 - log5) + log 28
= log(25*4/14*5 * 28)
= log 1000
= 3
To prove the given equation, we will simplify and manipulate the logarithmic expressions using the properties of logarithms.
Step 1: Simplify each of the logarithmic terms.
- log25 + log8 + log16/log64 - log14/5 + log28
Using the logarithmic property log(a) + log(b) = log(a * b):
- log(25 * 8) + log(16) - log(64) - log(14/5) + log(28)
Simplifying further:
- log(200) + log(16) - log(64) - log(14/5) + log(28)
Step 2: Use the logarithmic property log(a^b) = b * log(a) to simplify further.
- log(200) + 4 * log(2) - 6 * log(2) - log(14/5) + log(28)
Simplifying further:
- log(200) - 2 * log(2) - log(14/5) + log(28)
Step 3: Apply the logarithmic property log(a/b) = log(a) - log(b) to simplify further.
- log(200) - 2 * log(2) - (log(14) - log(5)) + log(28)
or
- log(200) - 2 * log(2) - log(14) + log(5) + log(28)
Step 4: Use the logarithmic property log(a * b) = log(a) + log(b) to simplify further.
- log(200) - log(2^2) - log(14) + log(5) + log(28)
Simplifying further:
- log(200) - log(4) - log(14) + log(5) + log(28)
Step 5: Use the logarithmic property log(a/b) = log(a) - log(b) to simplify further.
- log(200/4) - log(14) + log(5) + log(28)
or
- log(50) - log(14) + log(5) + log(28)
Step 6: Use the logarithmic property log(a * b) = log(a) + log(b) to simplify further.
- log(50/14) + log(5) + log(28)
or
- log(25/7) + log(5) + log(28)
Step 7: Apply the logarithmic property log(a/b) = log(a) - log(b) to simplify further.
- (log(25) - log(7)) + log(5) + log(28)
or
- log(25) + log(5) - log(7) + log(28)
Step 8: Use the logarithmic property log(a * b) = log(a) + log(b) to simplify further.
- log(25 * 5) + log(28) - log(7)
Simplifying further:
- log(125) + log(28) - log(7)
Step 9: Apply the logarithmic property log(a) + log(b) = log(a * b) to simplify further.
- log(125 * 28) - log(7)
Simplifying further:
- log(3500) - log(7)
Step 10: Use the logarithmic property log(a/b) = log(a) - log(b) to simplify further.
- log(3500/7)
or
- log(500)
Step 11: Finally, evaluate the logarithm.
- log(500) ≈ 2.69897
Since the simplified result is not equal to 3, the provided equation is not true.