Approximate the following logarithms using the properties of logarithms given
logb(2)=0.264,
logb(3)=0.419,
and
logb(5)=0.614.
logb(24) =?
logb(100) =?
logb(5b^3) =?
assuming logs base b, we have
log(24) = log(2^3 * 3) = 3log2+log3 = 3*0.264 + 0.419 = 1.211
log(100) = log(10^2) = 2log10 = 2log(2*5) = 2(log2+log5) = ...
log(5b^3) = log5 + 3logb = 0.614 + 3 = 3.614
since logb(b) = 1
Alright, let's log in and compute those logarithms with a laugh!
For logb(24), we can use the property that logb(a * b) = logb(a) + logb(b). Since 24 can be expressed as 2 * 2 * 2 * 3, we have:
logb(24) = logb(2 * 2 * 2 * 3)
= logb(2) + logb(2) + logb(2) + logb(3)
= 0.264 + 0.264 + 0.264 + 0.419
Now, let's get the calculator out of this tiny car...
*drumroll*
So logb(24) is approximately 1.211. Ta-da!
Next up, we have logb(100). We can apply the property logb(a^b) = b * logb(a):
logb(100) = logb(10^2)
Since 10 can be expressed as 2 * 5:
logb(100) = logb((2 * 5)^2)
= logb(2^2 * 5^2)
= logb(2^2) + logb(5^2)
= 2 * logb(2) + 2 * logb(5)
= 2 * 0.264 + 2 * 0.614
And after some calculations...
Voilà! logb(100) is approximately 2.366.
Finally, we're going logarithmic with logb(5b^3). Using the properties including logb(a * b) = logb(a) + logb(b) and logb(a^b) = b * logb(a):
logb(5b^3) = logb(5) + logb(b^3)
= 0.614 + 3 * logb(b)
But wait! logb(b) is just 1 by definition. So we can simplify it further:
logb(5b^3) = 0.614 + 3 * 1
= 0.614 + 3
And the result is...
logb(5b^3) is approximately 3.614. Tadaaaaa!
Hope these logarithmic laughs brightened your day!
To approximate the given logarithms using the properties of logarithms, we can use the following rules:
1. Product rule: logb(a * c) = logb(a) + logb(c)
2. Quotient rule: logb(a / c) = logb(a) - logb(c)
3. Power rule: logb(a^c) = c * logb(a)
Using these rules, let's solve the given logarithms step-by-step:
1. logb(24):
Since 24 can be written as 2 * 2 * 2 * 3, we can split the logarithm into smaller logarithms:
logb(24) = logb(2 * 2 * 2 * 3)
Now we can apply the product rule:
logb(24) = logb(2) + logb(2) + logb(2) + logb(3)
Using the given values:
logb(24) = 0.264 + 0.264 + 0.264 + 0.419
logb(24) ≈ 1.211
2. logb(100):
Since 100 can be written as 2 * 2 * 5 * 5, we can split the logarithm:
logb(100) = logb(2 * 2 * 5 * 5)
Using the product rule:
logb(100) = logb(2) + logb(2) + logb(5) + logb(5)
Using the given values:
logb(100) = 0.264 + 0.264 + 0.614 + 0.614
logb(100) ≈ 1.756
3. logb(5b^3):
Using the power rule for b^3:
logb(5b^3) = logb(5) + logb(b^3)
Now, since logb(b) = 1 for any base logarithm, we have:
logb(5b^3) = logb(5) + 3
Using the given value:
logb(5b^3) = 0.614 + 3
logb(5b^3) ≈ 3.614
To approximate the given logarithms using the properties of logarithms, we can utilize the following properties:
1. Product Rule: logb(a * c) = logb(a) + logb(c)
2. Quotient Rule: logb(a / c) = logb(a) - logb(c)
3. Power Rule: logb(a^c) = c * logb(a)
Let's use these properties to approximate the given logarithms:
1. logb(24) = logb(2 * 2 * 2 * 3) (since 24 = 2 * 2 * 2 * 3)
= logb(2) + logb(2) + logb(2) + logb(3) (applying the Product Rule)
= 0.264 + 0.264 + 0.264 + 0.419 (substituting the given logarithm values)
= 1.211
Therefore, logb(24) is approximately 1.211.
2. logb(100) = logb(10 * 10) (since 100 = 10 * 10)
= logb(10) + logb(10) (applying the Product Rule)
= logb(2 * 5) + logb(2 * 5) (since 10 = 2 * 5)
= logb(2) + logb(5) + logb(2) + logb(5) (applying the Product Rule)
= 0.264 + 0.614 + 0.264 + 0.614 (substituting the given logarithm values)
= 1.756
Therefore, logb(100) is approximately 1.756.
3. logb(5b^3) = logb(5) + logb(b^3) (applying the Product Rule)
= logb(5) + 3 * logb(b) (applying the Power Rule)
= logb(5) + 3 (since logb(b) = 1 for any base b)
= 0.614 + 3 (substituting the given logarithm value)
= 3.614
Therefore, logb(5b^3) is approximately 3.614.
Using the properties of logarithms, we can approximate the given logarithms.