Properties of logarithms |
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Property 1 |
loga1 = 0
Recall that if a ≠0, a0 = 1 therefore, loga1 = 0
Hence, the logarithm of 1 to any base is 0.
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| Example |
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log21 = 0; because in the exponential equation
we know that 20 = 1
log1/21 = 0; because in the exponential equation
we know that (½)0 = 1
The logarithm of unit to any non–zero base is zero ( where unity = 1). |
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Property 2 |
logaa = 1
Recall that in exponents a1 = a ⇒logaa =1
(where a ≠0, a > 0) |
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| Example |
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log1010 = 1 because in the exponential equation
we know that 101 = 10
log77 = 1 because in the exponential equation
we know that 71 = 7
logae = 1 because in the exponential equation
we know that e1= e
The logarithm of any non–zero positive number to the same base is unity. |
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Property 3 |
logaax = x
Recall that ax = ax |
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| Example |
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| Since you know that 34 = 34, you can write the logarithm equation as log334 = 4 |
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Try these questions |
| Find the following |
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1. |
log381
Answer: log381 = log334 = 4 |
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2. |
log61
Answer: log61 = x
6x = 1 ⇒ x = 0 |
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3. |
log1x = 23 Find x
Answer: log1x = 23 ⇒ 123 = 1
∴ x = 1 |
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4. |
log2256
Answer: log2256 = log228 = 8 |
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5. |
log525
Answer: log525 = log552 = 2 |
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6. |
If logx8 = 3 then find x
Answer: logx8 = 3 ⇒ x3 = 8
x3 = 23
x = 2 |
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